I would not advise doing this just for a nail trim. Is the vet experienced with birds or an avian specialist? If not, that may be why they are making that recommendation.
Alfie was gassed by a vet once so they could help with his beak (it was growing longer on one side and causing the beak to grow out wonky). They gassed him and sorted the beak but Alfie was sick after the procedure and very dazed/out of it from the gas for a good few hours. I found another vet who is further away but obviously has a lot more experience with birds, as he is able to handle Alfie a lot better, with a lot less stress and most importantly- no gas. I have never been back to that first vet and always travel the extra distance to see the vet that I trust.
I would personally seek a second opinion from another vet who is used to handling/treating birds.
The risk and stress isn't worth it for a nail trim. It should be achievable with an experience hand and a towel.
If you have any doubts about the care being provided by a vet then you are better to walk away and find another vet. It's always important to be able to trust the person giving care to your beloved pets.