Good Morning, Afternoon, Evening, Night!

Morning

Vinny we have a stove top fan got it from Aldi that seems efficient so I don't see why your plan wont work and its easier than the alternative of opening up the wall between  both rooms.

I have a stove top fan also. The woodburner is in the dining room which is laminatd .The living room is carpeted and I didn't want to be dragging logs in so there is a an open flamed gas fire in there. With gas prices going berserk lately I try not to use the fire in there much.The stove top fan helps to warm up the kitchen as well which is open plan but there is a door, a passage and another door to living room so very little heat percolates.
I spend mist of my time close to the woodburner, and Kato being a Siberian Husky spends most of his time in the living room where its cooler.
I know that air extraction is an exact science and it doesn't always do what you think it might, but at 20 squid, I'll take the chance!

I fitted a Victorian type fire in the living room which necessitated widening the chimney breast. There is a void of about 5 inches on each side.One side has heating pipes running up it but the other has nothing. My intention is to feed the fan into that void and with a flexi hose feed it out lower down out of the side of the chimney breast. This way it will not be noticeable in the living room