Ditto about the medication question above. Also, some breeders put leg bands on their birds, so that would need to be removed. If you are breeding the zebra finches yourself, you would need to make sure there is no medication exposure and also I would advise humanely euthanizing the bird first (via use of CO2 in a flooded gas chamber). The CO2 method is almost instintaneous with birds and would be more humane than feeding it to him live.
Lastly, if you are going to feed the pre-killed finch to the BP, you might want to consider clipping off the end of the nails and beak to help prevent puncture wounds.
Note also that feeding birds often result in runnier and stinkier poo.
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