A few things, after re-reading this:
"soaked the rat overnight"- in the refrigerator, I hope? If at room temperature that long, it was likely spoiled when you offered it. Bleah.
"we also tried feeding him when he was active at night so that he would be trying to hunt! but he didn't eat."- No surprise. BPs are "ambush hunters"- they do NOT hunt actively, they hide & wait to pounce on prey going by (to "get the drop on their prey"- aka by a "sneak attack"). Best to wait & offer meal when he's peeking out of his hide in evening/night hours, but never when he's roaming the cage actively.
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