The Wigner Function is simply a particular representation of a quantum state and so it only has an entropy in so far as the state does. One can ask are there any entropic quantities that have an elegant representation in terms of the Wigner function, and there may well be such quantities. Indeed, the linear entropy $1-\mathrm{tr}(\rho^{2})$ where $\mathrm{tr}(\rho^{2}) \propto \int W (p,q)^{2}$ has a neat form. However, like the von Neumann entropy this will give zero for a pure state! You already stated that you would like an entropy that does not give zero always so that you can make interesting statements like the above uncertainty relation. However, uncertainty relations crop up when you sum two or more entropy quantities.

I will expand on my remarks a bit more formally. The classical entropies, like the Shannon entropy, are defined on bit strings. We can define a quantum mechanical entropy by defining a measurement that gives us a bit string. For an observable $M$ with eigenvalues $\lambda_{j}$ and projectors $P_{j}$ onto the corresponding subspace, we can define a bit string $X_{M}(\rho)=\{ x_{1}, x_{2},... x_{j}... \}$ where $x_{j}=\mathrm{tr} ( \rho P_{j} )$. Now we can covert this into an entropy by classical means such as taking the Shannon entropy $S( X_{M}(\rho))$. If the state is an eigenstate of the measurement basis then the entropy will be zero.

How does the von Neumann entropy fit into this picture. Well an equivalent definition to the usual one is the following:
$S_{vonN}(\rho)=\min \{ S( X_{M}(\rho)) |M=M^{\dagger} \}$
which is simply the minimum possible measurement entropy.

Where do uncertainly relations come in? Well to have an uncertainty relation we must have 2 measurement observables that do not commute. If they have no common eigenstates an uncertainty relations follows by simply adding the 2 entropies. The inequality you have cited is simply $S_{P}(\rho)+S_{X}(\rho)$ for position plus momentum uncertainty. As noted in the comments this inequality can be saturated and so there is no hope of improving on it.

My opinion is that it is meaningless to ask for an entropy outside of a measurement context, and so this is what you need to decide on first. If it really is just position and momentum your interested in then I think the cited inequality says all there is!

This is my first attempt at an answer on stack exchange so I hope it is useful!

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