12,000 btu is a standard 3kw portable a/c these units are not that great to be honest, they work quite well to keep a standard size room cool in the summer
and this is in the UK lol its not that hot here, 90f outside in the summer
putting 4kw of light in a room , during summer and sealing the room with just a 12,000 btu does not sound good to me
one important thing to remember about a/c is that it recirculates air .. in order for ac to be efficient rooms are sealed so the air can be cooled
if you do not bring in new air directly from outside, or use co2, the air inside the sealed space will become dead co2 depleted air , this will cause lots of problems
so you still need to bring new air in .. which means the cold air from the a/c is getting sucked out, making it inefficient especially when a 12k ac uses 3kw
you could increase the size of your outlet and inlet fans, bring in air directly from outside that would help lower temps during summer
you could possibly run the ac and the extractor and inlet all on a control module so they work to allow new air to be brought in, then the extractor can turn off while the air is cooled , when that air becomes depleted of co2, the fans switch back on and bring new air in again
i have seen some folk, put ducting on the ac and feed it directly into the lights, this seems like an interesting idea , the extractor will suck from one end as the ac pushes cold air directly through the lights .. i think this could be the best option for a small ac
the cost of running ac, always makes me look for other solutions first
peace