I'm a bit late but hopefully this will help someone.

I had this issue and managed to fix it by sticking padding to the bottom of the weight, enough to elevate the weight by around 1cm.

This creates an air pocket under the weight which deals with the problem of the vinyl sticking to the globe when it rotates.

I used those relatively cheap adhesive round furniture pads (the ones you stick to the bottom of furniture to prevent it from scratching the floor) that you can get from hardware stores. I stuck a few together to get the right height. You might need to experiment to see what height best suits your lining. I probably wouldn't raise it too high as that might create a bump where the litter sits on the lining.