Sounds like "Fantasy-Slavery 2.0" ^^ but I generally like this idea. The thing is, you need a goal for the player and a reason why he behaves like this.

I really like the part "If the gnow wins the fight, the player gets money to invest in new items for his gnome (like prostitutes and alcohol)" I think it is always good to think out of the box If you add a small Tamagotchi/Spore flavor in it (the gnome grows dependent how you treated him and how he succeeded in life/the arena etc.), this could be really fun.

You can also build houses / loveshacks for your gnomes so that they can find partners, make some whoopee and get children which can help you as well. So, you could add as well some sort of "DNA processing" when two gnomes get a child with behaviour inherited by their parents and family tree.

Great idea!