One problem with feeding live is that many snakes get bitten in the process, & their owners aren't always aware of it. It might be only a bloodless pinch, but still even enough to cause your snake to hesitate (fear!) the live prey. If this is the case with yours, you might try feeding a younger rat (eyes closed) to see if that emboldens your snake. Using a live harmless* "appetizer" rodent is also one way to get a snake to learn to accept pre-killed rodents. Some snakes will get all pumped up eating a small live rodent, & as long as the 2 in combination aren't an excessively large meal, the snake may accept a pre-killed rodent immediately after taking (finishing) the small live one.
*When rats have eyes open, they fight back, & they can injure or even kill a snake that isn't interested in eating (for whatever reason, fear or illness, etc) so it's important to figure out what's going on- & yes, we highly recommend feeding pre-killed (fresh or f/t), but we also understand it can be very difficult, especially with BPs, because they rely on their heat sensing pits to recognize live rodents. When feeding f/t, are you aware of techniques to fool your snake into accepting a dead rodent? (heat & subtle motion) You have to get it just right, but maybe you just haven't tried hard enough to make it work.
Another possibility is that you're just feeding him too much & he's not hungry- this chart may help you assess the size & frequency of prey you're offering:
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