Two little updates now that I have the new Pi in my build.
No more lagging at all that I can discern. Whether the extra0.2GHz made the difference or there is another hardware issue with the other Pi is still unknown.
I figured out the undervolt issue, and it's something I think should be brought to people's attention for future troubleshooting issue with this type of build.
The WaveShare Game Hat I am using has a separate MicroUSB port for charging the 18650 battery you put in it. It seems that this port is NOT meant to EVER power the Pi directly under any circumstances, and cannot transfer full power from it. But because it completes the power circuit, it tries to do exactly that. To run this build on ANYTHING other than the onboard 18650 battery, the Pi's normal microUSB must be used.
Why I didn't think to test that sooner, I've no idea. I'm sure it was not suggested here by any of you who tried to help simply because you were unaware of the device's additional microUSB port - not everyone knows about this device, after all.
The good news for this is that it means I can continue to use the power bank I had been using before - it has an on/off button, allowing it to function as a switch for the Pi, and when plugged into the proper port, there is no undervolt warning and everything runs fine.
I thank you for your assistance, even though we were unable to find the specific issue with the previous board. Perhaps after I build my arcade cabinet with it, something else will come up that will lead us to solve it. But given what the cabinet will do and what kind of games were causing the lag, I don't think we'll be bumping into the issue anytime soon.