I grew in virtually the same space my very first time...
I think I have a link to the journal somewhere here....
Ah yes, here it is...
https://www.rollitup.org/indoor-growing/16572-out-old-new-grow.html
If you check the last page of that one it has a link to the flowering part of it also....
I started by building a cabinet inside the cabinet. I did this so that I could create a light sealed growing space. So inside the closet door it would open up to a sealed box and you had to remove the door to the box.
Reasons I did this, I wante to create a 3x2 space to grow in. I used the other two feet OUTSIDE the box to hold my carbon filter and fans and just ran the ducting into the room. By doing this I could have the other (left) closet door open so that it could vent the hot air out. I used the window right next to the closet to pull air in for the intake and hid the entire operation behind my TV Stand.
My roommate had no idea that I grew MJ in there until WAY after it was all done. He was quite impressed with how "stealthy" it was.
My suggestion and my experience in that grow is that you find an air cooled light or use CFL's. I used a 250wHPS (non-air cooled) that had an internal ballast. The thing got pretty damn hot. I ended up using a portable A/C inside the left side of the closet to pull colder intake air and keep things cool. That allowed me to lower my 250w light to about 3" and that made the buds bigger and tighter. BUT right now I'm vegging in a little bit larger closet, but I'm growing on a 2x4 tray with a 6 bulbs T5 fixture and I'm not having any problems keeping that cool.
So the way I would do it is to buy some T5's, some fans and get that running and cool for veg. Then switch to the bigger dogs and run an A/C or find some serious ventilation for flower.
I was able to get about 4 oz's in 8 weeks total (1 week veg, 7 weeks flower) in that setup. Worked for me back then and I had no complaints.