Can’t open the windows much as our cat is an indoor cat.

If you don’t have or want to get some sort of air conditioning, maybe obtain and stick a window screen on the window, then have the fan blow through that?

Not as common in the UK — here in the US, they were necessary to deal with malaria back before that got eradicated and kinda stayed for the convenience — but it looks like there are UK-based vendors and manufacturers, and that they’re often called “fly screens”.

You could even use something with larger-diameter holes, like chicken wire, if all you care about is the cat, but I figure that if you’re going to go to the trouble of putting something up, might as well block insects too.