The BI will actually be easier to deal with, as it will happily chow down on a variety of feeders, including chicks, mice, rabbit kits, etc. They don't have to ever eat rats.
Unfortunately you may have to rehome your BP's as it sounds like your allergies are worsening with time and repeated exposures. My only suggestion there is to go to the doctor, get tested for the exact allergen, and see if there is a course of treatment to mitigate your reactions. If not you could ask for a prescription for an epi-pen, just in case the inhaler doesn't work.
Also have you tried oral OTC antihistamines, taken an hour before you start warming the rats? Benadryl is usually the go-to but since it causes drowsiness I would try something different.
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