LAA permit
Englishal
I may be wrong but I'm pretty sure you can apply for a CAA permit for any aircraft. However, it's far more expensive as different rules apply. There are homebuilds on CAA permits (usually as they fall outside the LAA rules on engine size ect), so anyone could if they so wished fly their aircraft and not be an LAA member.
Personally I'm pretty happy with the LAA except that it does these days seem to lean slightly too heavily toward kit rather than plans built support.
SS