Welcome to Reef2Reef!
Stick around a while, many of us have or have kept herps, and you might just like the tanks we keep ;-)
I have never kept an Axolotl, and I've never owned or ran a chiller, so please double check all my advice here. But my thought would be to get the larger chiller. It seems obvious that you need a larger chiller if your current unit can't keep up. Remember, those tank size guidlines are just suggestions. Your environment (such as your poor AC situation) might require a larger or smaller chiller than typically suggested for your tank size.
And it's wise to be concerned with over cooling your guy (though from fish to reptiles to humans, overheating is a WAY larger problem than underheating, hands down). You may be wise to try a controller for your chiller. Inkbird makes an aquarium controller that *I believe* would work for you, and prevent your chiller from over chilling your tank.
InkBird Heater/Chiller Controller from Amazon
And in the short term, consider floating some frozen bottles of water in the tank, just to keep him cool during the hottest part of the day.
I hope that helps, good luck!