No as a parent you have to provide the necessities food, shelter, clothing. Anything beyond that, cell phone, TV, music, going out...... that's all removable until they take a shower. If it was my house all those fun things would be removed for a week for each day that a shower was refused. There has to be some punishment for the defiance or it will just get worse.
Um, not legally. It sounds reasonable, though. When you are dealing with children who don't want to bathe, you need to take action. I wouldn't say "starve" the child. Just tell them they are grounded, or can't play a game until they do. My little brother HATES taking showers.