What is a good product to keep bugs off screens that is safe for cats?

No pesticide is safe for cats, short of having a lot of spiders that are natural safe pesticides.

(Spiders are not insects but arachnoids and natural insecticides).

I’m allergic to chemical pesticides myself.

And I’m also allergic to mosquito bites (I get a rash and itch for 2 weeks, I was born in Paris, no mosquitoes there and I never got the immune system for them).

I have arachnophobia (fear of spiders) but I don’t bother them if they don’t bother me.

They stay in their corners, catching crawling bugs in their webs on the floor and flying bugs in their webs on the ceiling.

I make sure to not vacuum spider webs so my house can never look clean.

Between my cats and the spiders, I’m pretty safe from bugs, rodents and wild rabbits.

You might have an issue with your screen.

Maybe you have an aluminum metal one with bigger holes and should replace it with a fiberglass mesh one with tiny holes.

Take your screen to a store like Home Depot or Lowes and compare it to rolls of fiberglass mesh ones as it is NOT normal to have bugs get into your house through your screens that are supposed to keep the bugs away.

Once, I was given the stupidest advice about a flower keeping flies away in a house with no screen while on vacation in the South of France.

I put a vase of those flowers at the window (maybe lavender) and ended up with hundreds or thousands of flies inside my bedroom as they would fly in but never fly out because of the plant.

Once I removed the vase, all the flies flew out.

Mint, strong smelling herbs, and citronella are all supposedly pest deterrents. Not for all pests though. In their natural form, they aren't harmful to cats. Maybe try mixing some mint essential oil with water and spraying it on the screen? I don't know what kind of pests you've got, so it might not work but it's worth a try.