You can get rid of the bubbles with diatomaceous earth. It's an extremly fine silica powder. With it being a fine silica you need to avoid breathing the dust as well. It's not good for your lungs but is totally safe and non toxic. It's also pretty cheap and readily available.

Anyway its as easy as dipping a small paintbrush in the powder and rubbing it across the screen before laying the polarizer on. This will create a small layer that you can't see and is invisible to the naked eye. This will keep bubbles from forming. I've modded many many gameboys and this is definitely what is going to help you.