Is there anyway to stop a bmx bike brake wearing down the paint on the rim?

No.

BMX bikes have rim brakes, they work by pinching the rim between two pads. The surface of the rim will wear away eventually, there's not really any way around that. Best you can do is keep the pads and rims clean from any embedded grit.

If you're really stubborn, you could have a BMX modified to take disc brake wheels, but it'd take some serious metal working skills.

Or you can replace the rear wheel with a wheel that has a coaster braked hub.