I need to pressurize a room and I'm kind of at a loss as to how to do it, hoping someone out there might have some advice
It depends if you're looking for a "proper" solution or any workable solution, and your tolerance to noise.

It looks like filtering is one of your requirements, so you'll probably want to use a squirrel cage type fan, which is capable of developing the requisite pressure to push air past a filter.

Common sources of these fans: 
-Air purifiers (also filters the air for you), 100-600CFM depending on what you get, often cheap on Craigslist or Facebook Marketplace. Might be too small. ~25-200W. Also try "air scrubbers", available with higher CFMs like furnace blowers.

-Furnace blowers or old furnaces (cut the furnace part off, leave it, or just build a box) and filter, 2000-6000CFM. ~300-1000W. Usually has a few speeds. Also try "air makeup unit". Not toooo loud.

-Bathroom fans. Broan LoSone (pretty big)(300cfm) ~250W, Costco has some real quiet ones ~100cfm last I checked ~50W.

-Air mover blowers for drying floors or whatnot (need to rig a filter onto this), pretty cheap at Home Depot ~100-300w.

-Range hood fans - get a "chimmney style" one which will likely be squirrel cage and replace the grease catch filters with a furnace filter. Often cheap on Craigslist etc. ~200-400W.

-Hack a filter into a portable air conditioner or salvage the fan(take the "condenser" fan)

-Search Craigslist for words like "fan" or "blower" and find something that looks like it takes at least a few hundred watts

Crappy options:
-Box fans with furnace filters on them
-Induced draft blowers, small squirrel cage blowers, pressure blowers etc etc are a little too small for the room I think

Expensive options: 
-Proper air handler with filter
-in line duct fans (unusually expensive for some reason)

Before you commit to installing something, try an experiment first: Tape up and seal the fan with filter rigged on to one of the doors with plastic or something, and make sure the plastic bulges out significantly.