Is there a common notation of the power set excluding the empty set?

I am not aware of any such notation (and in the business of choice functions, one runs a lot into $\mathcal P(S)\setminus\{\varnothing\}$).

It is fine to make your own, but be sure to be consistent about it, and to define it at the beginning of your work.

There is a risk of having too many notations, it may burden the reader. Sometimes just writing it explicitly works just as well. If you're tired of doing that, write a LaTeX macro.

I would suggest using something close to standard notation, e.g. $\mathcal{P}_{\lt \omega}(X)$ or $[X]^{\lt \omega}$ is used for the set of finite subsets of $X$, so why not $\mathcal{P}_{\geq 1}(X)$ or $[X]^{\geq 1}$?