One of the court-approved visitation superivisors finally called me back. She said that there are just three of them in the county, and they each charge $50 per hour. For that, she meets you at a public park, spies on your interactions with the kid, and files a report to the court.
To get on her schedule, you have to first have to do an intake interview with her and sign a bunch of contracts. So does the other parent. Currently she has a waiting list, and other two supervisors are also very busy.