I just had to have one at one point and with even more stringent limitations you describe so instead of the entry level 375, mine was a 9.3x74 in a Chapuis. I found in double guns not only the name and model determines price but also what it's chambered in-those in the big classic cartridge generally cost quite a bit more. Even going to the 375 H&H or 375 R jumped the price up a bit.
The Chapuis was probably at the bottom of your dollar range at about 4K and certainly not in a preferred dangerous game cartridge though, no doubt, it's been used that way.
Anyway, it was well made and a whole lot of fun to shoot; took a running buffalo heifer with it at about eighty yards.
