What is the advantage of using Bessel function as the basis in a Hankel Transform?
The Hankel Transform includes the Bessel function as part of its definition. I wouldn’t say the Bessel has an advantage. I would say that the Hankel Transform has an advantage where a two dimensional signal has radial symmetry, or saying it another way, the signal in $x$ and $y$ can be expressed as a product of two functions in $r$ and  $\theta$ through a change of variables.
$x=r\cos \theta$ $y=r\sin \theta$

$$f(x,y)dx dy \Rightarrow g(r) h( \theta) r dr d \theta $$

The Hankel Transform is equivalent to a Fourier Transform on the radius $r$ variable. 

Any advantage over STFT or Wigner would need to be explained in the literature you read.  Im not aware of any general use of Hankel grams, or why the Hankel would make any obvious relevance outside of two dimensional geometries with rotational symmetry.