Another option would be to bypass the flash altogether. I have made a flood for my security system using slightly longer wavelength IREDs which are totally invisible to me, unlike the faint red glow from the lights built into the camera. The camera has a somewhat lower response to these IREDs but it works.
Assuming that your game camera has a shutter also the flood concept makes it really simple, just leave it on all the time. If you can't afford the power drain you can just substitute the flood for the flash.