Suppose I am a member of a club/group. What is that state of being inside the club called, in the following context?
I have __ to this club.
I'm thinking admission, but that really doesn't sound like it.
I have membership of this club. (BrE)
I have membership in this club. (AmE)
Considering the way the pattern is specified, I'd probably go with access or admittance, as in "I have access or admittance to the club". Nonetheless, in almost any other case I'd consider membership as the best word to describe the fact that someone pertains to a club or organization.
The original phrasing of the question differs a lot from your suggested idea of a better term than admission in an example sentence.
'What is that state of being inside the club called...?'
The answer to 'being a member of the club' can be 'belong', as in:
I belong to this club.
(This can be a powerful statement, say, when someone questions your locus standi within the club premises.)
However, if you must use the form 'I have __ to this club.' only, then you can say:
I have access to this club.