Is aquarium temperature really that much of an issue for you? I live in the UK and we don't have hot summers or AC, but my aquariums that are typically 24/25c (75/77f) can get upto 30c (86f) for a week or two in the summer without any problems. This has happened the last 2 summers when we had record breaking temperatures.
Most heaters work with a thermostat, so they automatically go off when the water reaches their set temperature. So turning off a heater wont make any difference if its thermostatically controlled.
Lights will heat up an aquarium a little, so turning off the lights will lower aquarium temperature more than turning off the heater.
Removing the lid and running a fan over the surface will promote evaporation and reduce aquarium temperature. But you don't want to do this because of your cat. We have a cat too.
There are aquarium chillers, but they are expensive. Thats really the only way to get an aquarium temperature below whatever your room temperature is at. So controlling room temperature is the better way to go. Close curtains and windows during the day, open windows at night to promote nighttime cooling. Turn off any appliances that don't need to be on. Maybe consider getting a small evaporative cooler for your bedroom.
The biggest problem with warmer water is that it holds less oxygen than cooler water. So make sure your water is well aerated to maximise oxygen content if the temperature gets higher than normal.
I wouldnt try using bottled frozen water to control water. Ive heard some people do this, but all you are going to do is create temperature swings that are going to be more hazardous than just letting the water naturally settle to the room temperature when its warm.