"Wouldn't a reactor on Mars have different cooling issues to one on the Moon? Sure, there's not a lot of air on Mars, but what's there is sufficient to make cooling a reactor a different proposition to cooling one on the Moon. And a shorter diurnal cycle means a different thermal regime as well."

I would assume the reactor would be in the polar regions of the Moon and maybe in a dark crater- and perhaps the reactor part buried in any case.

You probably are going to design it so the reactor would not meltdown if for whatever reason the coolant failed- it might not kill anyone if it melts down but you lose something worth a few billion dollars- and of course deprive anyone of any power it's providing.
So, the reactor itself may not have much difference between the Moon and Mars [or on Earth]. The difference with earth is you will be installing it in a vacuum [or on Mars, a near vacuum]. If you putting a reactor on Mars, probably your entire mission will be dependent on having no problem with getting it to work. Testing that one in fact would likely have an almost certainly of having no problems might be a good idea- 5 days out rather than 5 months out.

In addition you aren't going to build one or two reactors for Mars- you going to build some and test a few- and maybe one or two will end up in some museum or the other. If one is having a nuclear reactor being on the critical path to a successful Manned Mars, then it could make sense to build them early and test one on the Moon.
And if you plan to deploy the nuclear reactor robotically- it seems to me it's all the more reason one do it on the Moon first.

Anyways about the difference of the cooling affects of 1/100th Earth atmosphere as compared near zero. I guess it depends on how much of that 1/100th Earth atmosphere, you plan force over the area you want to cool, big fans/or even jet engines could give you a lot of cooling.
But if you just talking about passive cooling, I don't think it's a significant difference. I mean a human body in a space suit on the Moon or Mars will generally need to be cooled, though one will need less cooling on Mars, but I would think whether one is on cold or hot ground or in sunlight or shade would be bigger factor than the amount of atmosphere.

On Mars one will have the wind and that would have a significant affect, but the wind probably isn't predictable or dependable.