Wht you do not want to do is convert a vented gas logs setup into an unvented one. You want a screen that will keep the bees out, but not have a bad effect on air flow, so that fumes from the fire will still get out.

If I were you, I would contact a certified chimney sweep or a gas logs onstaller/contractor and explain your problem and ask for a free estimate to fix it. If he agrees to do that, you can ask him to explain what exactly he would do, and you can decide for yourself whether or not to do it yourself.

Decided not to use any screen, the screen size that was permissible would not keep out the insects.