I would definitely NOT go over the border to see a rheumy. I would take Advil or something similar in the meantime to deal with the pain. You mentioned you play a lot of sports, well you may have to temper that for the time being unless the Advil relieves the pain completely. If that is the case, then I would continue to play. I was on an NSAID for a long time before my arthritis got really bad and I continued to play badminton and work out in the gym. A couple of months is not long to wait if you are not in too much pain. I'm a Canadian too so I understand your frustration, but once you have your first appointment with a rheumy, you will find that after that, the next appointments will not be a problem. It is the first one that is hard to get. Honestly, why go to the States when you are going to get the same treatment for nothing here. If you were in excruciating pain, I'd tell you to contact your doctor and then he would put you on the high need list or at the very least, cancellation list. You could actually do that now if you want. You could call the rheumy and ask if there is a cancellation between now and your appointment, would they please call you. The other option is if you are in excruciating pain then you could always go to emergency and they would do all the tests right there and then BUT only if you are in excruciating pain. Otherwise, you are misusing the system. Hang in there, I know it is hard but honestly, it probably wouldn't make a bit of difference and you would have wasted $500 just to be told you RA. Plus, you will have to be followed by a doctor here anyway.
3 months can seem like an eternity when you are waiting to see a specialist. It is possible that you do have RA. Early treatment is important but I am not sure what you will buy by coming to the states. I will use my own case as an example time line.
After a year and a half of going to different docs trying to get a diagnosis that felt right, I finally got a referral to a new rheumy. I had to wait 4 weeks to see him and he ran more tests then another 4 weeks to get those results and a laundry list of problems with me.
Total time from onset of symptoms to treatment with a drug for RA was 2 years. Cost of doctors appointments and tests was thousands with insurance. Without insurance it would have been insane. I had x rays, lab work, MRIs and scans. An MRI by itself can be 3-5 grand depending on how long the test is, whether they use contrast, etc.
Because you are able to function and can see a doc for free in 3 months and get all your tests for free at the same time, I would recommend you stay on the path you are on in Canada. If you run into problems getting diagnosed and treated in Canada, then I would come south for treatment, but be prepared for the hefty price tags that come with health care here and investigate it in advance.