Welcome to TFP. One of the moderators will likely ask that you fill-out your profile so people can understand your pool, equipment and needs better. I suggest that you do that and I also suggest that you consider maintaining your own pool with guidance from this site.
I wasn't aware of Lanai Accessories before and I looked at their site. They offer two styles of clamp mounts and one style of screw mount but they don't give much information on the items. Why don't their mounts fit your uprights? Might you have the wrong style or a different width of upright?
There is a huge variety of "pool pole mounts" available, just do a Google search. However, all I've seen besides the Lanai Accessories ones are screw mount that would require drilling small holes into your uprights. This should be easy and probably more secure than just clamping. If you are in Florida as I expect, I'd suggest using metal mounts/hooks, plastic gets weak, brittle and discolored in the intense sunshine, even within the cage.