Just my two cents:
- fans at bottom, leave the rest open.
- This setup (four inches space between stove and wall is meant as fire precaution. Any restriction of free airflow and you made a hot chamber behind all the decorative finish. It is compromise between aesthetics and functionality.
EDIT - i didn´t read its a gas stove, so never mind, it has probably some sort of fire protection...
- wire protection covers (salvaged from PSU or bought separately - something like that:
https://www.amazon.com/Easycargo-Gri...FXDP14FFV3CMAY) do not restrict airflow and add ton of finish.
I fabbed something similar for my parents wood stove with four 12-cm PC fans and no thermostat, just PWM speed control module (cca $3.90, e-bay) to controll it manually. (Unfortunatelly my mother is more for aesthetics and does not like to mix "country" with "techno", as she said)
If you want to go redneck style, get temperature controlled switch from old electric kettle. Closes/opens at boiling point.
Sounds like nice project to me!
P.S: If you want no project, just 100% style, go for this:
https://www.ecofan.co.uk/
or this:
https://www.stirlingengine.co.uk/d.a...VULCANSTOVEFAN
(there are also cheaper e-bay versions, but afaik these are the originals)
- you could try thermocouple modules to power your CPU fans... No need to control, runs when your stove is hot enough... - get inspired by that ecofan