Carbon filter first grow

So here's my problem. I'm growing in a 2x4x5 grow tent in my small basement. I'm using led lights and t5 side lighting. It's winter so the basement stays at 60 degrees F. My lights keep the grow tent around 80 degrees but if I turn on my exhaust fan with carbon filter it gradually brings the tent down to 60 degrees. I have a speed controller for fan and it still brings the temp down. The basements probably 10x6x6 so could I just mount the fan and carbon filter above the tent on the outside so it cleans the air without sacrificing the temperature inside the tent. I'm trying to grow 8 small plants so I'm guessing it's going to smell pretty bad so trying to figure this out now before they start to smell.

Thanks

just recirculate with the filter instead of exhaust it out. and you would only need to run the filter the last 3 weeks when it starts to put out smell

The real problem that I found with smell, is that it only really occurs at the end of bloom, and during drying. Check it out by not running the fan, and see how strong the odor is. It shouldn't be too bad as long as the plants are not disturbed. Personnaly, with the temps you are mentioning, I would opt for the 80's, and forget the charcoal until it is really needed, say at week 6 or 7 of bloom. Good luck