There seems to be two ways teams approach the problem of getting stuck on rings. You can either build your robot with high enough clearance that rings can just pass through between the two sides of your drive under your bot. Or alternatively you can build the drive low riding enough so that rings just get pushed, not slip under and get stuck. But as far as I know there is no way to guarantee you’ll never get stuck on a ring, it just seems like something that happens if your bot lands on one in just the wrong way.
as far as using mecanums go, see this post discussing their traction in comparison to omni wheels: Omni wheel vs Mecanum - #13 by technik3k
to summarize it, mecanum wheels have significantly less traction than omni wheels both on foam tiles and on a plastic surface like the platform. This doesn’t mean mecanum wheels won’t be able to climb the platform, that depends on a number of factors, not just traction, but it might mean that you’ll have a harder time climbing than if you were to use omni wheels.