A small spray bottle filled with rubbing alcohol makes a fine 'portable shower'.

You can spray your armpits and give yourself a light going-over. Also turn your bike clothes inside out and spray them and hang them up to dry.

The locker room, and you, smells like alcohol for a bit. But that's not a big deal.

I know this doesn't specifically answer your question, OP. But it's a whole lot simpler and it'll work anywhere.