No, there were no prescribed flags for cossack regiments in the field. The Ataman Regiment had a flag granted to it at the end of the Napoleonic Wars, but no other regiment got anything officially until later in the 19th century. Like regular Russian regiments, cossacks probably had various icons and religious banners available, but if they were regularly seen on the battlefield I would have to say, "Prove it." They had more religious significance than military.
The flags described in Viskovatov are single items given to the Don Host as a whole, and I have no idication that they ever left the Don Host home territory.
For what it's worth, here's the Viskovatov text:
In 1803 this host was granted a flag with a light skyblue center and dark skyblue edges, surrounded in gold. In the middle, from each side, was a gold cross with a spear and shaft, and silver eight-pointed stars. On the edges was the inscription in gold: “By the Grace of God We Alexander I Emperor and Autocrat of All the Russias, Moscow, Kiev, Vladimir, and Novgorod, Tsar of Kazan, Tsar of Astrakhan, Tsar of Siberia, Tsar of the Tauric Chersonese, Lord of Pskov and Grand Prince of Smolensk, Lithuania, Volhynia, and Podolia, Duke of Estland, Livland, Courland, and Semigalia, Samogitia, Karelia, and Tver, Yugor, Perm, Vyatka, Bolgaria, and others, Lord and Grand Prince of Novgorod of the Lower Land, Chernigov, Ryazan, Polotsk, Rostov, Yaroslav, Belozersk, Udorsk, Obdorsk, Kondia, Vitebsk, Mstislav, and all the northern lands Ruler and Sovereign of the Iveria, Kartalia, Georgia, and Kabarda lands, of the Circassian and Mountain Princes and others the Hereditary Sovereign and Lord, Heir of Norway, Duke of Schleswig-Holstein, Stormarn, Ditmarsen, and Oldenburg, Sovereign of Ever, etc., etc., etc., grant this flag in place of that given in 1706 and renewed in 1746.” On the reverse side: “To the Ataman and cossacks of Our Host, for their many and loyal services, especially that undertaken in 1705 during the Astrakhan troubles, in eternal and immortal memory, in the year of our Lord 1803, and of Our Reign the 4th year.” The fringe, cords, and tassels of this flag were silver; the spearhead with Emperor Alexander’s monogram in the center was gold; the pole was green, and the flag cloth was sewn to a crimson cloth wound around the pole (Illus. 2490).
In the same year of 1803 there was granted an all-white flag with the State coat of arms surrounded by the arms of Moscow, Kiev, Vladimir, Novgorod, Kazan, and Astrakhan, and by silver and gold wreaths and flowers. Around the flag, between panels outlined in gold, was the same inscription and above, but after the words “etc., etc., etc.” was “granted this flag in place of that given in 1722 to the loyally subject Don Host for their service rendered during the war with the Swedes and for the permanent peace concluded with the Swedish crown. The year of our Lord 1803, and of Our Reign the third.” The fringe, spearhead, and all other apurtenences of this flag were exactly the same as for the above flag (Illus. 2491).
In the same year of 1803 there was a white flag similar to the preceding except with very insignificant differences in the images and the addition of four silver crosses on a blue field, surrounded in gold, In the inscription, after the words “to the loyally subject Don Host,” there followed: “for their service shown in creating a permanent peace with His Majesty the Shah of Persia. In the year of our Lord 1803, and of Our reign the third.” Around the silver crosses on a blue field were the word in gold: “Christ we glory in Your cross and sing Your Resurrection and glory that You are our God in the Holy Glorious Trinity.” (Illus. 2492). This flag was granted to replace that received by the Don Host from Empress Anna Ioannovna in 1733.
In the same year of 1803 there was a white flag with, in the center, a two-headed eagle on whose breast was a dark-blue shield edged in gold, with the gold monogram of Emperor Alexander I. Above the wings alongside the crown in double gold circles were gold crosses, and between the outer and inner circles were words in gold, written in Church Slavonic characters: “Christ we glory in Your cross and sing Your Resurrection and glory that You are our God in the Holy Glorious Trinity.” Around the flag, between gold stripes, was the gold inscription: “This flag is granted in place of that given in 1764 to the loyally subject Don Host for their service rendered during the late Prussian war, 1803, of the reign of His Imperial Majesty, the All-Merciful Sovereign Emperor Alexander Pavlovich, All-Russian Autocrat, the 3rd year.” The fringe, spearhead, and other appurtenences of this flag were the same as for the above (Illus. 2493).
In the same year of 1803 the Don Host was given a white standard [znachek]with a two-headed Russian eagle, the gold script monogram of Emperor Alexander, and various decorations. This guidon’s fringe was silver, and all other appurtenences were the same as for the above flags (Illus. 2494).
In the same year of 1803 the Don Host was given two standards [bunchuki] of a pale straw color with raspberry edges. On it, between gold stripes, were gold and silver stars. On one standard was the archangel Michael in a rectangle, defeating the devil, and on the other, in the same kind of rectangle, was a silver cross surrounded by silver stars, and in the upper corner nearest to the pole, a hand coming out of the clouds. The fringe and other appurtenence were the same as for the above flags (Illus. 2495).
In the same year of 1803 the town of Novocherkassk [capital of the Don Host lands-M.C.] was granted a coat of arms that was made up in gold and silver and with paints, on a rectangular dark-blue cloth surrounded by gold edging. It was fastened to a crossbar that hung by a cord below the spearhead of a normal flag pole. In the upper part of the cloth, above the coat of arms, was the gold inscription: “Coat of arms of the Don Host town of Cherkassk, in Graciously granted by the Sovereign Emperor Alexander the First, on the 3rd day of October of the year 1803.” Below, under the coat of arms, was a description of the arms in gold letters: “A shield, divided cross fashion into four parts, with a gold top in which is visible half of a black, two-headed, crowned eagle as it flies away; under this eagle on a red field in cross fashion: a gold mace [pernach] and two kinds of cossack scepter-like symbols of authority—a naseka and a bobylev khvost, and below, in a skyblue field – another kind of mace [bulava], and two cossack symbols of authority—a bunchuk and another naseka; in the side parts, on a silver field, on the right side, are laid four flags crosswise: two white, one skyblue, and one red, with black eagles depicted on them, the flags being tied together in the middle with a laurel wreath, and on the left side above a river is a red fortress.” The cloth on the crossbar, the fringe, cords, tassels, and vertical pole were the same as for the above flags, while the spearhead was gilt with an Imperial crown on top and the Imperial monogram in the center (Illus. 2496).
In 1807 Khanzhenkov’s and Sysoev’s Don Cossack regiments were each granted a white flag. On one side was a two-headed Russian eagle in an orange circle surrounded by gold wreaths, with a gold crown on top. On the other side in gold was the monogram of Emperor Alexander I in the middle of a gold sunburst. On each side of the flags was the gold inscription: “For the feat at Schoengraben 4 November 1805, in a battle of a 5-thousand strong corps against 30 thousand of the enemy.” This flag had a gold fringe; silver tassels with an admixture of black and gold silk, hanging from St.-George ribbons; gild spearhead, with the cross of the order of St. George in the middle; green pole (of the pattern for standard poles in the Regular Cavalry, with gold longtitudinal stripes (Illus. 2497).
In 1811 the Don Host was given a white flag with gold monograms, wreaths, and crowns in the corners, and with semicircles of St.-George ribbon on which in silver was the inscription: “To the loyally subject Don Host for service rendered during the campaign against the French in 1807.” In the center of the flag, on one side, was a gold cross on a silver field surrounded by gold wreaths and flowers, and on the other—a two-headed Russian eagle on a gold field surrouned by gold wreaths and flowers. Gold fringe, silver cords and tassels; gold spearhead with an Imperial monogram of the same in the center; pole as for the preceding flags (Illus. 2498).
In 1814 the Ataman Regiment was given a dark-blue flag with a painted image of the Savior surrounded by the gold inscription: “God is with us, loosen your tongues and abase yourselves, for God is with us.” Around the flag is another gold inscription: “Awaken, Lord, Your mercy toward us, for we worship Thee, yea without fail for eternity” and “To the Don Host Ataman Regiment for bravery.” This flag’s fringe was gold; the spearhead and tassels were as for the 1807 flag described above; green pole with gold longitudinal stripes (Illus. 2499).
In the same year the same regiment received a white standard [bunchuk] with an image of St. George slaying a dragon and surrounded by the gold inscription: “To the Don Host Ataman Regiment for distinguished courage.” This standard’s fringe was gold, while the spearhead and tassels were as for the abov flags (Illus. 2500).
In 1816 Dyachkin’s regiment was given a green flag with gold monograms in the corners, on a white field with gold crowns and wreaths. On one side of the flag, in an orange cirlce surrounded by gold wreaths, under a likewise gold crown, was a Russian two-headed eagle, and on the other side, in the same circle, was a crimson cross with gold edges and in a gold sunburst. On each side was the gold inscription: “To Dyachkin’s brave Don regiment.” The flag’s fringe was gold; silver cords and tassels; gold spearhead, with a St.-George cross (Illus. 2501).