Learn English – Does the Oakland Athletics logo misuse the apostrophe

You are correct about the punctuation.

First of all, using ' and not 's is typically only done when there is a possessive of a plural subject:

The two cars' engines.
The many Christmas trees' lights.

(In the past, it used to be a stylistic convention to put only a ' after a singular subject that ended in s. Although that is still done by some people, the convention has shifted away from that recently.)

The subject in question here seems to be The Bad Boys Club.

Despite having the plural boys as part of its name, it is a singular club.

There are two ways of interpreting this:

1. The Bad Boys Club's T-shirt.

Here, it's a single T-shirt that is owned by the club.

2. The Bad Boys Club T-shirt.

Here, it's a type of T-shirt that represents The Bad Boys Club. (The club name acts adjectivally.)

This is similar to wearing a Nike shoe rather than a Nike's shoe.

But in whichever interpretation, the sole apostrophe after Club isn't correct.