Either Big Sur or Monterey and using OpenCore Legacy Patcher. Make sure you have a data backup, a bootable USB installer with High Sierra for emergency rollback, and know what newer hardware you will purchase as plan C if things go really sideways.
I think Big Sur was when the OpenCore project finally managed to get working acceleration patches for the HD 6750M and other AMD TeraScale GPUs. I don't know if that fix was backward compatible with Mojave/Catalina.
Monterey is newer, but has a couple spots that are broken without Metal (see known issues:
https://github.com/dortania/OpenCore-Legacy-Patcher/issues/108)
Ventura/Sonoma start to accumulate known AVX2 issues and probably aren't worth installing if you are just trying to get Catalina+ to install a mainstream browser update.
Or you can do like I did and just stay on 10.12/10.13 to have a dedicated machine for legacy 32-bit Mac Apps.