I have the same exact setup, here is what I did. Cut hole top of closet, vent wtih PC fan (12 volt) use any 12 volt charger cable to power up PC fan ( i use a charger cable i had laying around that charged up my cordless hand held vacumm - dirt devil), 4" dryer duct - home depot,connect one end to bathroom exhaust AFTER bathroom exhaust fan very important the other i have a panty hose over end of duct slip on a 4" sheet metal coupling and connect to the top of closet outlet.
Thats how i set it up, PC fan sucks exhaust out my closet into discharge duct of bathroom exhaust DOWNSTREAM of bathroom fan, that way bathroom fam does not have to be on for it to work connect your ductwork from the closet as close to exit of the house.
1. You wont create blow back when bathroom exhaust is on.
2. When you do use bath room exhaust fan it will create whats known in my business a venturi action.
I know what i speak of i'm a retired HVAC tech ( heating, ventilation and air conditioning).
If you dont have a nearby exhaust you have plumbing vents to connect to.
This is for a small grow nothing big,
2-3 plants max. Room where closet is located has central air so air is positve pressure goes under closet door into closet or crack open door for more air if needed.
Doing NL, AK 47 and 48 grow.
This is my first grow. But I think i'll do ok did tons of research first im confident mistakes are going to be kept at a minimum but i will make mistakes, of that im sure.
Good luck!