Yes, I remember that dilemma when I took my very new Speedy for a day in London!
I think I put him between my knees.
Now the handles are dark and I just put Speedy anywhere.
I do clean him regularly - I read a magazine that said you have nasty e.coli type germs on bag bottoms from toilets.
If there is a hook in the loo I'll try and put the bag on there.
Worst places - train toilets!!! Ugh.
I carry a plastic bag and put my LV on that...