A lot of companies sell vents you can install in your air box. You drill out the required size hole and the little mesh screens pop in. But In your case I think the holes are a little big. So I think if you cleaned up around the holes real nice and just got some screen for a window and cut out a square to cover the hole from the inside, you could glue it in place with some silicone or epoxy of some sort. It will help keep out the extra debris.
I just removed my lid, I have too much water and mud where I live to do anything else to the air box.
The holes are 1", my original plan was to use those vents. They said they to drill 1" holes for them. Where I drilled the holes the air still has to go thru the filter so I didn't buy them. But yeah it would've been the same as just taking off the lid like you.
Same here with the mud that's why I bought a outerwear for it. Thinking about making some sort of snorkel kit for it just to get the filter up higher in the air.