my bathroom sink burps

I don't think the screw on vent cap is the issue because it's a one way only allowing air into the tank and preventing it from escaping the tank. It sounds like you have a problem with the vent that goes up to the roof. Most likely you're going to have to get up on the roof and remove the cap over the vent. Test the shower with the cap removed, it's possible that the vent pipe was left too long and is blocked closed by the vent cap. If no change in the behavior the pipe might be plugged. If you have a willing helper have them crawl under the rv and listen while you drop a few ice cubes down the pipe and see if they hit the bottom. Alternatively a heavy fishing weight and some heavy test line can be lowered down the vent and see if you hear it hitting the bottom of the tank.