Wood stoves throw out a LOT of radiant heat. If your tank is on a straight line of sight from the stove, it will warm considerably when the fire burns and cool when the fire goes out. If you could somehow shield the tank, that would make a big difference...
I lived in a house about 6 years ago that we heated primarily with a wood stove... the house was small and pretty open. Objects in the same room as the fire would get toasty warm, as will your fish. If you don't think so, stand in front of a fully heated wood stove. Then have someone stand between you and the stove. You'll be amazed at how much cooler you feel.
If you can block the radiant (i.e., line-of-sight) heat, you should be OK. The changes in room temp should be handled OK by your heater.
Good luck,
Jim