What would be correct “Businessman’s club” or “Businessmen’s club”?

Neither is more correct than the other. They're both common words. In fact, they're sometimes different words. For instance, to oblige someone is to do something for them that they want you to do. "She asked me to pick up dinner, and I obliged her by getting some lasagna from a little Italian place down the street."

That's not the same as obligating someone, which means to make them HAVE to do something (or feel that they have to). "Although the statue was meant as a birthday gift, he obviously felt obligated to get me something expensive in return, despite my protestations."

There is a dialect aspect to it as well. What's wrong with "obligate"? In US English nothing is wrong with it. In Australian English everything is wrong with it. Here's how I'd have put it.

"Although the statue was meant as a birthday gift, he obviously felt obliged to get me something expensive in return, despite my protest."

In my dialect "oblige" is fairly common but "obligate" is never heard. The distinction that Uriel mentions is not made in Australian English. We use "oblige" for both meanings.

According to Wikipedia:

In British usage, some words of French, Latin, or Greek origin end with a consonant followed by -re, with the -re unstressed and pronounced /ər/. Most of these words have the ending -er in the United States. The difference is most common for words ending -bre or -tre: British spellings centre, goitre, kilometre, litre, lustre, mitre, nitre, reconnoitre, saltpetre, spectre, theatre, titre, calibre, fibre, sabre, and sombre all have -er in American spelling.

... Many words spelled with -re in Modern French are spelled with -er in both British and American usage; among these are chapter, December, diameter, perimeter, disaster, enter, filter, letter, member, minister, monster, October, November, number, oyster, parameter, powder, proper, September, sober, and tender.

The ending -cre, as in acre, lucre, massacre, mediocre, is preserved in American English, to indicate the c is pronounced /k/ rather than /s/.

After other consonants, there are not many -re endings ...: louvre and manoeuvre after -v; meagre (but not eager) and ogre after -g; and euchre, ochre, and sepulchre after -ch. In the United States, ogre and euchre are standard; manoeuvre and sepulchre are usually spelled as maneuver and sepulcher; ...

The e preceding the r is retained in American-derived forms of nouns and verbs, for example, fibers, reconnoitered, centering, ... fibres, reconnoitred and centring respectively in British usage. ...

Exceptions. ... include Germanic words like anger, mother, timber, water and Romance words like danger, quarter, river.

...

More recent French loanwords retain an -re spelling in American English. ... double-entendre, genre, or oeuvre. ... cadre, macabre, maître d', Notre Dame, piastre, and timbre.