I've been through the whole myco thing with my rats before(its the whole reason I'm so insistant about quarantining all new animals). Is it possible you can take your rats to the vet to get some meds? I know its an off and on thing, but myco can get pretty nasty.

To be on the safe side, I wouldn't house a bird in the same room as rats that have myco. Plus, if you plan on letting the bird have time out of its cage, then there is the possibility that it could land on the rat cage. A rat's bite on the leg of a small bird like a budgie(or any bird for that matter) would be a tragedy.

Good luck in your decision. =D