Re: So basically....
"am I misunderstanding how they work"
No, you're right. Heat rejection is a big problem for the moon. I don't know how deep you'd have to dig for a "lunathermal" solution, either. On earth it might be 20 feet or so, depending [then you just put a bunch of water piping undeground to reject heat into the earth]. On the moon you could do the same thing, but it's not very 'portable'.
I assume that the ginormous circular reflector is for that purpose. NASA knows how to make this work, as they did it on the space shuttle. But it's worth pointing out that any heat engine (including the Stirling engine) needs a differential temperature to work. So does a peltiere device, which is what I originally thought they'd use. But peltier devices aren't as efficient and would probably weigh more for kilowatt capable power.
I would also expect inverters and batteries to manage short-term transients. I can't see speed control on a Stirling engine being all that easy. If differential temperature drives it, then that's also what controls it [and you can't heat/cool things fast enough for speed control to work really well]. So they'd probably use 'load control' to also control speed, and use batteries as a 'surge' to keep speed within a safe band [with some safeties to burn off power or 'put on the brakes' when necessary].
In other words, controlling a Stirling engine generator could be like balancing a pole on your nose. Then again it might have some natural speed control built in, due to fluid friction and/or momentum, with that back/forth fluid motion thing going on.
It's a fair bet that the stirling engine + nuke plant would respond poorly to rapid transients, though. But the U.S. Navy solved this a long time ago in their own power polants, since you can go from "all stop" to "all ahead balls to the walls" in a reasonably short time on THOSE systems. Even so, electric power generation transients are usually a LOT slower than "flooring it" on a propulsion system. So maybe it's not an issue?