The first-person shooters I’m about to recommend aren’t necessarily the most popular or well-known, requiring you test them out a bit to see if they are to your liking, OP. Here goes:
Eradicator is an interesting “2.5D” first-person shooter for MS-DOS that gives you a slightly less conventional arsenal of sci-fi guns and a selection of unique characters with their own quirks. It has no issues running smoothly even on older pre-1995 Pentium systems (Pentium 90, in my case), with or without pipeline-burst cache that you get on Intel Triton-based chipset motherboards.
The Terminator: SkyNET is a fairly early polygonal 3D-graphics FPS with both on-foot and vehicle-based (i.e. “Halo style”) multiplayer. Might be interesting if you’re looking to try something more asymmetrically balanced, with the whole humans vs. robots theme going on. If you use the 320x200 video mode, it should be able to run pretty well on any Triton-based system. At the higher SVGA/640x480 resolution, on the other hand, I suspect that you will need a CPU closer to 200 MHz to claw back any lost frames (can’t speak from experience, sadly).
Outlaws is another pretty cool “2.5D” FPS that should be able to run well at 320x240 on any Triton-era system. At 640x480, a Pentium Pro or MMX processor should ideally be used (in my own experience). This is even more so the case if you insist on using A3D-accelerated audio, which Jedi Knight also supports.
The games below are not FPS games, so they may not be your cup of tea, OP. They are games that shooter fans might still enjoy, so here goes:
Interstate ‘76 is plenty of fun if you like car-on-car weaponized combat and customization. You can also practice against computer opponents to get used to how the game works! It does require a Pentium that’s a fair bit beefier than what the box mentions (Pentium 90, minimum) to actually enjoy it (again, in my experience). However, my P90 system is one with a 430NX chipset (no pipeline-burst cache), meaning that a Triton board with speeds at 133MHz (or slightly north of that) should have a much better time, even if has to settle for 320x240 pixels. 640x480 requires 200 MHz processors to run at the “intended” speed (below 30 frames/sec. or thereabout) with higher detail settings.
Jazz Jackrabbit 2 could probably be lots of fun too if your party would rather try a side-scrolling platformer/run-and-gun shooter hybrid instead of 3D shooters. This game shouldn’t have any problems running fast on Pentium 1 systems at the lowest 320x240 resolution, which I wouldn’t recommend, seeing as how your peripheral view suffers, putting you at a great disadvantage (unless all players agree to use the same resolution). At higher resolutions, however, this is where you will need a fairly beefy Pentium (200+ MHz) to not bog down performance too much. The graphics are fairly customizeable, so it shouldn’t be too much trouble enjoying it on any Pentium, though.
Your friendly source of public-domain PC game video captures on period-correct hardware! Youtube: @dipshidian