I'd take the Fulbright, but that's just me. It'd only boost your qualifications, so I can't imagine they'd defund you. But that's just me.
Any way you could find out how likely it is for them to take your funding away?
Worst case scenario, apply to another school if they take your funding away. Don't GO there, but use their funding to prove to your 1st choice that they aren't your only option, and need to resurrect that competitive offer.
You don't want to have a shoulda coulda woulda scenario. Keep focusing on making yourself better while you're on your ETA, and there's no way they wont give funding when you return, because you'll be the bomb. They'll be lucky to have you.