You should be able to run WishList Member with a cache plugin as long as you avoid caching pages when a member is logged in.
Most cache plugins will have a setting for this. It will only deliver cached pages to non-logged in users.
If a member is logged in, it ignores the cache.
You also do not want to cache any registration pages.
To avoid any potential issues, most cache plugins have a method for you to exclude pages from their caching.
Adding the following WishList Member pages to that exclude area should prevent issues:
Please be sure to clear the cache in your cache plugin before any testing.