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, manufacturers, and installers, and that they're often called "fly screens" rather than "window 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.