Is a gas fireplace enough to heat a small cottage?
I'd talk to the Fireplace dealer/installer to confirm, but yes, that should work provided:


you use a blower (sounds like you have)
your house isn't chopped up into too many rooms
given the high ceilings, I'd strongly suggest installing a ceiling fan as well to help. If you don't, you may end up with one really hot room, and a bunch of only 'warm' rooms.