The license is very clear about it: you cannot use the license to publish character generation software, just like you couldn't use the d20 STL to do so. So no logo on your software, in short.
Whether you can do so
without the license has the same answer as every other such question in the history of the universe: it falls under normal copyright law, and you'll need to consult a lawyer if you're unsure. Nobody from
WotC will give you legal advice on that.
The fan site policy might present a third avennue for doing so, but you'll need to wait and see.