Another month and another update.
I did end up purchasing another CPU. I paired it with my new motherboard but ended up removing both form the build after a bit of a misadventure that ended in something smelling like smoke. No idea if I accidentally fried the CPU, the motherboard, or both, but decided it wasn't worth continuing tests with that particular combo for fear of potentially damaging more components.
Misadventure aside, I ran into a strange issue off the bat where after I installed the CPU I wasn't getting signal to the display. I ended up putting the other CPU back in and testing but got the same result. I reset CMOS but again no luck. I ended up reseating the RAM and that did the trick. I put the new CPU back in and it seemed to do okay at first but when I loaded up Battlefront I again lost signal. I left it for the night and when I reapproached the next day I was getting inconsistent behavior.
At this point I decided that the benefits of maintaining the Windows 98-XP dual environment was no longer worth the cost of dealing with all these issues. I decided I needed to start fresh to see if the extra memory and accompanying workarounds were a potential cause. I was actually in the middle of reinstalling XP when the system crashed and I smelled smoke.
With my original motherboard and CPU back in the case, I proceeded with a fresh install of XP with only one stick of RAM. XP installed without issue. I loaded up the drivers (much easier this time without having to negotiate two separate operating systems) and installed Battlefront again. Played for about forty minutes and, aside from Battlefront itself crashing, the system performed just fine. I tested it again this morning without issue.
I think it's too early to say whether I am out of the woods completely on this but I'm hopeful given the initial positive results. Since all the hardware testing seemed to show no glaring issues, I am persuaded that the problem ultimately sat with the the dual-boot environment I set up and the workarounds I implemented. I suppose it's also possible the system did work initially but became unstable over time with use.
If the build continues to work well with the fresh install, I think that will point to the above being the case. Although I am somewhat sad to no longer have 98, I feel better knowing that things are (currently) more stable. If I ever feel the need for an 98 machine again (and have the space), I'll make a separate machine. Glad to have accomplished the dual-boot at least once but I don't think I need to attempt it again.
Thanks everyone for their continued help and comments. I'll report back after a bit and let everyone know how things are going.