You should be able to mount the fan right to the filter and stand it in the corner out of the way. Just have it suck air thru the filter and blow it up to the ceiling.
It doesn't have to be up high if you have good air circulation in the room. Sure hot air rises but not when it's getting mixed up all the time.
I've been wanting to do a sealed room using a dual hose portable AC unit too but a bit short of cash. Picking up a few sheets of OSB next week and add on a veg space downstairs that can become part of the better setup down the line.