I'm going to chime in and say that I think it will be insignificant. I don't know for sure, but I really doubt you could do much to cool the stove. When it's cold in the Yukon, I'm literally stuffing in ~8-10 big logs every ~12 hours.
It's not at all uncommon for the base of the chimney to be glowing red. Obviously not ideal, but it happens.
When it's past -40, I set my alarm for 1am to get up and put in more wood. The stove would just last the night if I didn't, but it's a pain to light it again from scratch every morning, so it's easier to just keep it going.
All of that is to say the wood stove is absolutely pumping out heat 24x7 from ~September to ~April.
If I already have batteries and charge controllers and inverters, why not wire in a handful of TECs.. even if I only get a combined total of 100W, that's worth having over those months the sun is not up for long, and not strong.