Pentiums are meant to drag arse at 386 speed with all the caches turned off and set to lowest multi, so you can give that a try. Make little switches on 2 or 3 pin headers to put on a faceplate to plug into the jumpers so you can do it without taking the machine apart every time.

Thanks for the information! I think I can also use the 'setmul' utility, right?

Does that mean I should just keep the faster Pentium 3 (slot 1) machine? (ie. is there no benefit to having the slower Pentium 1?)

Sorry, I guess I was answering whether you needed another 486 machine, suggesting you try that on the slow pentium, to see if that gets you what you want. setmul only works on mobile and some other CPUs.

How can I find out if I upgrade the IBM 350 to a Pentium 233 MMX CPU?

The included CPU is labeled A80502166. The Wikipedia page for the IBM 350 says it included up to a Pentium 233 MMX?

Slot 1 P3 makes for a great dos/9x PC. You have an ISA slot for dos sound card and something like a P3 500 should be able to play just about any Win9x games that don't run well in XP.
The only benefit of keeping the the socket 7 PC is if playing any speed sensitive games.
https://www.vogonswiki.com/index.php/List_of_ … sensitive_games

Of course if you want a socket 7 PC then that's fair enough!
You can configure it with things like a voodoo 1 and nice sound card
and have the P3 more 9x focused with a later voodoo, alternative dos sound card and even set it up for some lan games (even if you've no one to play with)

I'd think your safe with the CPU upgrade.

486's don't really make for great retro PC's Too fast for early games but not fast enough for late dos games. Socket 7 is really a much better system.
That said I own 3 486's and no Socket 7's as have no nostalga for early Pentiums.
Even then all my dos games work just fine on my P3, its only when I want to go "full retro" with a pure dos/Win3x build do I fire up a 486.