There is no problem to start with - or a minor one. Make sure the cooling system is on the USV and possibly has a secondary USV giving it 5 minutes more and you are good. Done.
See, the moment power fails (and the USV runs out), the servers are out. This means you do not get new heat (because power is out), and only have to deal with residual heat. THis is parts of the server being higher temperature heating up the room. This is VERY little unless you have something like a glowing metal box in there. There is no need for cooling - temperatures may go up, but if you bother reading the manuals of all equipment, you may be shocked by the allowed temperature ranges.
So, temperature control on the UPPER side is fool proof, UNLESS you have serious environmental conditions (i.e. the warehouse is in death valley in the sun and the hall will heat up to 70 degree without active cooling, of you are in Norway in winter and things will freeze). There simply is no heat being pumped into the room to start with.
I'm concerned about heat in the server room if the equipment is running and
there is no cooling.
This can not happen. Period. Not if the cooling runs on the USV as it should.
THat said, there is another angle to your question. IF you establish enough passive cooling to cool the server room, then the active cooling JUST WASTED ENERGY AND MAINTENANCE COST. Because obviously the passive cooling is good enough to just use it.
Your main problem is having a good or secondary air con because if that one fails, that is a single point of failure and i.e. a pump or compressor failure on the air conditioning does NOT turn the additional heat from the servers off.
Consider using passive cooling with ventilators. ONLY. Moisture is basically only a problem if the equipment is COOLER than the warehouse so that moisture turns into water on the servers. As long as the servers run, they are more hot. Unless you have really hot air, you are much better just using multiple ventilators instead of air conditioning and cycling fresh air through the server room. Intel tested that IN THE DESERT - worked like a charm. Servers have no problem running in 35 or 50 degree celsius. Check the spec sheets.