I double checked the army lists for 1800 & 1802, to ensure that I had not made an error with the spelling as "Willet"
The name is listed as Ralph Willet Adye in the body of the lists under Royal Artillery and as Adye Ralph "Willett" in the index.
In 1798 he is listed as Ralph Willett Adye.
I also note that few officers had their middle name listed in full, Ralph being one of the exceptions. If someone used an 1804 list and the entry was similar to 1802, etc. I can see where they might think that Willet Adye was his last name?
The London Gazette notices of his promotions usually list as Ralph W. Adye.