Hi, have you considered window double exhaust fans?
Something like this:
DEF get the extenders to fully block the window. During the day, point both fans to blow out. Keep the blinds closed above the fan. In the evening, do one in one out, or both in. It will not let your cat out, it holds solidly in the window, blocking it fully and can't be just pushed out as it sits in the frame and cradles the window itself, it actually fits so tightly it's a pain to get OUT of the window lol. It can also deal during storms. I lived in unairconditioned apartments in Chicago when the heat was over 100, and while it didn't make it COOL it made it tolerable.
Drink tons of water. Cool showers in the evening before bed without fully drying off. Have a room fan to blow on you at night if possible. For really hot nights, I froze large water bottles and wrapped them in a tshirt and held them against my chest to fall asleep since it cooled my core down enough to fall asleep. Rattan bed mats can keep your bed a more comfortable temperature, too.
Consider getting cooling towels for out in public, large sunhats (they make ones designed for evaporative cooling properties), taking breaks when walking. Again, drink TONS of water. Being hydrated will let your body regulate temperatures more. Try to avoid doing a ton of activity during the middle of the day. Consider ventilated shirts. Even a parasol to keep the sun off of you - shade makes a big difference.
I'd also consider speaking to your psychiatrist if you are on any meds. Some psych meds can make the body significantly more heat intolerant. There might be some strategies for your specific meds if you are on any that have this as a side effect.
I hope at least some of this was useful.