You seem to be not getting along with your roommate, so I thought I would/might if you wanted to move back home.
What would a native speaker say?
You thought you would/might what?
(cross-posted with SD)
I am a Non-NEST (non-native English-speaking teacher)
Something is wrong with your sentence. You should have given the context. The only way I see to make your sentence sound correctly is this one:
You seem to be not getting along with your roommate, so I thought you might want to move back home.
If I were a native speaker of English, I would never shut up. :-)
'Would' is the natural choice.