Refinance first -- although your wife may have to be a co-borrower and/or be on title depending on the lender and other facts not stated. Whether or not your wife is "on title" she will be able to assert a community property interest in the house unless there is a clear waiver of that interest. IMO, a Quitclaim deed will not suffice. Good luck.
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You should likely consult a lender first. Lending requirements VARY greatly from lender to lender. Often (but not always), private mortgage brokers (i.e. not your local national bank chain) have greater flexibility in lending issues. Their underwriting department will ultimately decide if they can lend to you, and what (if anything) they may require regarding the trust, deeds, etc., etc. Good luck!