I've experienced the same thing. I think it's caused by setting up on the bench properly, ironically. Shoulder blades down/chest up puts your back muscles in a shortened position, then you contract them hard for the lift. I don't know if you ever get foot cramps, but same idea. Flex your foot hard and pretty soon it'll start to cramp.
I've had to foam roll before each set of BP, especially if I feel that the muscles are a bit "twingy". I'll also dig around with a lacrosse ball if needed and get usually get a nice pop from my vertebrae.
So, here's what's going on with me, you may or may not think it also applies to you. If you assess my spine, I have areas of relative hyper and hypo mobility. My whole spine is very stiff, but the movement comes from a few specific places. One of these places is T11 or T12 - where my T spine ends and the lumbar spine begins. The other place is around T5/T6.
Imagine you're setting up for a heavy OHP. You've got the weight in your hands and you lean back a bit. Ideally, that extension is coming from all of your vertebrae more or less equally. Instead, my spine "hinges" backwards at the 2 areas mentioned above. Those are actually areas of relative hypermobility.
So you're wondering how this relates to your issue...When I lay down for a BP, I set up and get some T extension. That extension is coming from T5/T6 (the same place you may be having problems, it's relatively common). The brain sees this relative hypermobility as instability and causes the large muscles around it to forcefully contract to protect the area of instability in the spine.
This is what happens to me, something similar might be going on with you, and this is why foam rolling and digging around with a lacrosse ball immediately before a set helps. It restores mobility to the entire spine, thereby decreasing the points of relative hypermobility.