Question: It hadn’t properly registered yet with Arthur that the council wanted to knock it down" - what's the meaning of register with someone here?

Answer: 
This would be dependent on whether the speaker was referring to the group/entity of the Club or to the location of Sunnyside Country Club.

For instance if you removed the words 'Sunnyside' and 'Country' and just left the word Club, a speaker would be more likely to choose the term "to". 

I was accepted for membership to the club.

One might then wonder as to which 'club' you are referring, so it is necessary for the speaker to use the Proper Name.

Once the proper name is used however, it also has the interchangeable reference to the location of Sunnyside Country Club. If the speaker was referring to the location they would more commonly use 'at' in this sentence.

I was accepted for membership at Sunnyside Country Club. (as in over at their location, where they meet.)
I would suggest this is more of a US v UK English issue. UK dictionaries use 'of' when defining membership. Moreover, I've lived in the UK for over 50 years and until about five years ago I had never heard the phrase 'membership to' - it was always 'membership of'. Suddenly, 'membership to' has become ubiquitous. As a Brit, 'membership to' makes no sense to me, but had I grown up in the USA it would doubtless seem normal.

I'd add that I'm an editor by profession, so I have a reasonable grasp of UK English and am very alert to changes in usage.