I'm not very knowledgeable at all, however, I feel that I can safely answer your question here without overstepping my authority.

Any tongue on the palate is better than no tongue on the palate. You say that the posterior tongue doesn't stay up, implying that you can keep it there for atleast a short while before tiring. Do it as much as you can. You will tire out your tongue and will have to rest it for a day or two sometimes, but it's strength will build and you will be able to keep it there longer and longer. Like any other muscle.

Another thing to mention is that you might experience alot of strange sensation in your tongue as you are training yourself to keep it on the palate. Like for example; pain, swollen tastebuds, tired tongue so it can't move certain directions, and just weird, unexpected stuff like that but it's all temporary. Just keep at it.

An exercise that would be perfect for you would be "Tongue chewing" from Mike Mew (Orthotropics); https://www.youtube.com/watch?v=ZNocCJNicrc&t=136s

Yes especially when I open my mouth a little bit I'm able to lift the posterior part of my tongue. Maybe I try to do this lifting exercise every day from now on to train the muscles. And when the muscles are trained enough I could try to do the lifting motion with my mouth closed.