Yes, the i7-6700K is compatible with a Gigabyte H310M A.
Yes, the i7-6700K is compatible with a Gigabyte H310M A.
I found a cheap CPU for you and wondered if it would fit your old PC. It has the same socket, but LGA 1151 was discontinued a long time ago and they've changed a lot about it. I checked the board listings but didn't see it there. Some people say it should work with a BIOS update. It might be worth trying that to get the 15 FPS you're aiming for with your i3 8100.
Technically you can adjust the BIOS of a 300 series board to support a 1151v1 CPU and vice versa, though this isn't recommended. It's not worth it when you find affordable CPUs on 300 series boards that already work without changes.