You could maybe just remove the mechs from lancer. Maybe like double the health pools of pilots so they fight longer

Go big and try Rifts. There's at least some way you could force the GM to have a cave oriented game in that one. Oh! Set it in Missouri and be ordered by the Coalition to find the fabled Cheese Caves! https://www.sfhscollegeprep.org/news-detail?pk=1237113

Or HERO System if you want to be heroic (and really crunch those systems)

It would be a stretch but I think Dawn Patrol could work.

Getting stronger while you play? Like with weighted dice?

Theres a cool game called Slugblaster which has a similar concept, except that you're going into alternate dimensions. I am sure you could handwave the dimensional travel as caveing

Check out Uncommon World for a wonderfully refined version!

Thanks so much! https://troypress.itch.io/uncommon-world for anybody else wanting to check it out

Appreciate you linking it!

Paranoia is always underground, you just need monsters.