You could try lowering your left stirrup leather to help mount easier? Not sure if that’s allowed or if you’d have time to put it back up, I’m not familiar with mounted games.
If you're going to learn mounted games, then the only way you can really mount during them in swinging up from the ground. This is a lot easier on smaller ponies, which is one of the reasons why they tend to get used more than horses. It is possible to learn to jump onto horses from the ground though. Jockeys do it all the time. It just takes time and patience. If you have access to a smaller (quiet) pony, it might be worth first learning on it.
When I need to mount and all the blocks are taken I just use the fence. That is just my way of doing so