In you all’s situation MFS likely makes the most sense; the only thing is she’d need to paper file as you don’t have an ITIN. You’re in one of the weird situations where MFJ might actually hurt because it would disqualify you from the foreign earned income exclusion (FEIE)
She’d paper file as MFS, manually write in “NRA” for your SSN, and mail it off to the appropriate IRS office. She would need to claim the FEIE and would likely owe $0 in taxes. I’d recommend mailing any back taxes in soon as paper can take a minute.
This is not tax advice, and if you have specific questions it’s worth consulting a CPA with experience in expat filing.