Would this work?

Native American Brothels.

Several years back casinos were frowned upon by most states. Then Native Americans slowly began to open casinos on reservations across the country with taxes going to the state. Soon many reservations were competing to own casinos as the profits generally helped the lifestyles of the reservation members through distribution of the profits and jobs for the reservation.
The reservation members benefited with distribution of profits and more jobs.
The reservation itself benefited with more money coming in.
The state benefited with increased tax revenues.

Allowing Native American reservations to own brothels would have the same effect as casinos. It would bring profits to the reservation, bring more jobs, and produce tax revenue for the state. So why not why not start pushing for legalized brothels on reservations? If casino's could slowly be accepted this way, why not brothels as well? This is of course assuming the there would be some sort of set of rules in place similar to Nevada's to ensure the safety of the brothel employees.

How did Native Americans start the push for casino's? How, and more importantly, would they start a push for brothels on their reservations?

Sadly it didn't work like that. Many states frown upon tribal casinos. The laws for gambling changed in the late 80's, early 90's. It was illegal to have casinos outside of Las Vegas and Atlantic City. When the laws did change it was mainly for "tribes" back east. Than slowly word traveled to California. The biggest state to not allow tribal casinos is Texas. Cause tribal casinos must be regulated by the state and federal government. And the revenue is split not as good as you think. I believe in Arizona the split is 85/15. State run casinos split is about 55/45.
Before a casino can go up, they must first settle on a gaming compact. This compact is to be decide by tribal, state, and federal governments to run the casinos.

The reason that a tribal brothel wouldn't work is it's against state and federal laws. If the laws did change than maybe, but it would have to go to a vote within the tribal government. So it really wouldn't work. Think of it this way. Say a tribe does approve a brothel. All that would happen is as soon as people visited the brothel, state police would wait til you get off tribal land and go to arrest.

Another thing. Tribes don't need brothels. They have casinos.