CZ 457 American.
My latebrother had a Grade II BLR. Very pretty, but only a so-so shooter. He sold it when he bought a Kimber 82.
For the use you describe, I’d be tempted to buy a Browning SA-22, which should come in right about $600, last I looked, unless you insist on a Belgian.
I agree with Pappy the SA 22 was built for rabbit hunting. Lighting fast and a dream to carry.