The "Regiment de Paris" was a provisional regiment formed from the first battalions of the two regiments of Garde Municipale de Paris (2 battalions each) by an order dated 30 November 1806. The provisional regiment (termed the Régiment de Paris) left Paris on 22 December, apparently marching to occupy Cassel.
After Eylau, Napoleon sent orders for the governor of Cassel to send the Regiment de Paris and the "regiment italien" to Berlin (to be replaced in Cassel with Dutch troops).
On 11 April N ordered the Regiment de Paris to join Lefebvre’s corps at Danzig.
On 5 May, N ordered the formation of Lannes' Reserve Corps. Verdier's division was to include 2e légère, 3e de ligne, 7e de ligne and the Regiment de Paris, but 2e legere and the Regiment de Paris were not to join until after the capture of Danzig (which did not occur until the 29th of May).
I don’t have any specifics on whether or not the regiment had rejoined Verdier’s division prior to Friedland and I have the same reference you have to the unit being in Mortier’s corps (Dupas’ division). Belhomme shows the Regiment de Paris with Lannes’ Corps as of 8 June 1807, but he is working from the formal organization and wouldn’t show detachments or on-the-spot adjustments. Interestingly, 2e légère had been left with the R de P at Danzig until the capitulation and presumably marched east with the R de P. But 2e légère shows up in OOBs with Verdier’s division at Friedland as originally ordered. Both regiments of the Paris Municiple Guard list Danzig and Friedland battle honors, so it seems the provisional regiment was definitely present at Friedland. It’s entirely possible that this regiment was reassigned from Verdier to Dupas at some point between 29 May at Danzig and 14 June at Friedland. Or the OOB showing it with Mortier rather than Lannes might be in error. Does the regiment appear in any of the accounts of the battle? I haven't read anything on Friedland for quite a while.
But I think that answers your original question, even though it produces another one!