Our catalyst-equipped stove really cooks, but everything I've read says you basically have to keep it hot enough for catalytic action or the catalyst will get clogged up.
I know a lot more about catalytic stoves than I do tractors, so we could go into quite a rabbit hole, here.

Basically no, you do not have to keep the stove above catalytic reburn temperature, you only need to ensure enough fuel (wood gas) is being fed into the catalytic combustor to keep
that above reburn temperature, usually right around 500F. It's common for me to run my stoves down around 250F stove top and chimney, but the combustor inside is still cruising at 500F.
Basic principle of operation is to get the whole stove up to 500F while running with catalyst bypassed, to get the fire going well and the catalytic combustor (aka "cat") pre-heated to near "light-off" temperature. Then you close bypass damper to force exhaust through the cat, which thanks to some special heavy-metals coatings will "light off", burning any of the volatile wood gas products remaining in the exhaust.
After you achieve light off, you can lower the burn rate of the stove way down to the threshold of just providing enough combustion byproducts to the cat to keep it active. Many of the better catalytic stoves can turn down so far that they appear to be not burning at all, like a picture box with barely-smouldering or even apparently un-lit wood sitting in it. How far you can actually turn down is very dependent on how dry the wood is, as less-dry wood can tend to burn out when you try to smoulder it very low, and evaporation of any water remaining in the wood is a cooling process that can lower combustor temperature below 500F.
The real advantage of the cat stoves is that the best ones can achieve better than 10 hours burn time per cubic foot of firebox, whereas non-cats are generally limited to 4 hours or less per cubic foot of firebox. The trade-off is that you're not exactly getting huge BTU's/hour at those very low burn rates, but often that's what you want to avoid overheating the house.