the shed on the photo can't really host more than a 250W hps and that will already produce a lot of heat.

You will also need an exhaust fan / extractor to renew the air and get as much of the heat out

If you can order some led that would be much better.

It will be hard to use efficiently the A/C but if it's powerfull enough to cool the room from where the shed suck the air it could help.