I am sorry he is having a problem that needs the help of a rheumatologist at all - esp at his young age. Are the joints too mobile or not mobile enough, do you know? Or did she say they just were not moving properly? I ask because I had to see a rheumatologist for years and even though I was misdiagnosed the exercises and therapy I did made it so that I retained a lot of mobility in my joints that most with my problems lose early on. Regardless of the specific problems, unless he is extremely hypermobile, range of motion exercises done on a regular basis now can make a big difference in his adult life.
What the doctor wants him to do is physical therapy. Some PT places have lap pools, at least here. Others just have smaller pools for stretching, aerobics, etc.... You will need to call around. If there is not a PT place with a lap pool, are there exercises he could do in a smaller pool that would help? The doctor needs to write a rx for PT that states what she wants him to do and also includes an evaluation. Often docs do not have a real clue what the PT (and/or Occupational Therapist (OT) if that would help, likely not in this situation) can do, so you will need to ask the PT if there are other things that can also help.
If there is a YMCA in your area, they WILL let him swim. Most likely for free after you call and explain that you cannot afford membership but have a child who needs to swim regularly to treat a problem as recommended/prescribed by his specialist. They may want a letter from the doctor (both to see that you are truthful and to be sure he is healthy enough to swim and it won't hurt him). You will likely need to be there to supervise. They may not have open swim times, but if classes are not using the entire pool or if it is adult swim time and he will mostly leave them alone (under your supervision), they will usually let you use the facilities.
They don't advertise this kind of thing, largely because a ton of people would take advantage of them, but I haven't heard of one that is unwilling to help in this kind of situation. When the docs said swimming would help thank you with Occupational Therapist (OT) stuff and help his muscles work up stamina (he couldn't do exercises, PE, etc... without tiring much sooner than his peers - part of the Sensory Integration Disorder (SID) problems according to the Occupational Therapist (OT) and PT people we saw) - our Y let us come in so he could do exercises and walk in the shallow end of the pool while the adult exercise and lap swim was going on as long as an adult was in the water with him.
If you have an indoor public pool run by your community chances are that they will also allow you to use the facilities for free or a much lower cost. Otherwise medicaid MUST work with you to provide the care he needs. It won't be easy if there isn't a contracted PT place that has a pool, but they should.
DON"T FORGET to search the website AND to call them to see if they pay for transportation. Here they do NOT come out an tell you this, NOT EVER unless you say something to them about not being able to get to to the doctor or whatever because you don't have a car, but they have a transportation program. With notice they will provide rides to and from an appointment or the pharmacy or they will give you an authorization number nad will reimburse you for your mileage. 24 hours notice is needed for most rides, but authorization numbers can be given the day of the appointment, though they like you to call ahead a day or two.
With regular therapy needed, esp if you have to go to a nearby town, this could really add up.
If the doctor doesn't want to do a PT referral, you will have to push her. Usually docs would give this to you on a prescription form during the appointment but the doctor may not be aware that if it isn't then medicaid won't cover it.
There is no reason why medicaid would NOT cover this, as long as the doctor says it has to be done.