How much space is there outside the door? The reason Im asking is because I think you should use roll out shelving.
Buy two 2'x4'x8' rolling metal shelves and put your plants and lights on those. You said you had 4 - 250 watt lights < This should be enough to get you started this time but you may want to upgrade to 400's next time or add 2 more 250's and add in another shelve in the unit.
But since you have 4 lights you need 3 shelves per shelving unit and and two lights per shelving unit.
Since you only have 250's and very limited space I would keep them small like maybe a ounce per plant if you use 1 gallon grow bags or pots you should be able to fit 8 plants per shelf giving you 32 plants or if your plants yield an ounce per plant - 2 lbs. NOTE the picture is with 4 shelves and 3 lights per unit you will be using 3 shelves and 2 lights per unit
I dont know if your ballasts are built in to your lights or if they are separate but if they are separate you can stick them on the top shelf. Then you can zip tie all the cords to one of the corners going down.
Also And I didnt put this in the picture MAKE SURE TO HAVE SOMETHING TO CATCH ANY WATER!!!! the last thing you want is your lights underneath to get wet!!!
YOU ARE GOING TO NEED SOME VERY GOOD VENTILATION AND A COLD AIR INTAKE if its still to hot in there maybe a portable ac unit? I think heat will be the biggest problem.
So what I would do is once you have your clones rooted I would stick them straight into flowering. Then when you need to water the plants unplug the lights and pull the shelving units out the door and into the open room. Water them and make sure no water has gotten on any lights or can get on them then push them back into the closet and plug them back in. I would get a thermometer for in there 2 so you know if its getting to hot in there and I would do all the clones at once so you dont have to move 2 much around. Another thing to think about is reflective insulation on the walls this should help by keeping the walls from absorbing heat
Smell -- I have used this method and found it works pretty good
https://www.rollitup.org/grow-room-design-setup/63004-my-3-carbon-filter.html
Also use good soil, dont feed them to much because they will be small and wont need much extra. And let them go for longer in flowering than it says for them to go (1-3 weeks) because they will keep on adding weight and its easy to pick to early but hard to pick to late.
Anyways the think that covers most of the bases. Hope I Helped and Happy Growing!
If you decide to do this idea I want that 20 $