I know what you mean, I had it today while cycling, BTW I can't claim to be anywhere near to fit and I know what caused it for me, but I would say if your GP says you shouldn't be running right now then you need to stop. You are doing other excersises that will keep you in good shape, could you find something else more suitable to replace the running in the short term?
I assume RA is Rheumatoid Arthritis? Is your remission induced by steroid? That might be significant, as steroid use can affect blood sugars, and the symptoms you're describing sounds like they could be related to blood sugar. If you're going to carry on running, I'd consider taking some isotonic drink with you and eating a high starch food - maybe a banana, no longer than an hour before you start and as soon as you finish, to try and keep your sugars more regulated.
I was heading for a cool down yesterday! Problem was I was out with OH and DCs and when I turned off he headed up a (small) hill and DCs followed. I thought, wrongly as it turned, that I may as well follow them and get a drink later. And we're in the southern hemisphere, it was near on 30 degrees C. Doh!
Running 5k in 25 mins isn't slow, that's fairly hard going IMO. I run 5k in about 29 mins and feel like throwing up when I'm finished as well as dizzy etc for a minute or two. I'm also near max heart rate in the last stretch and probably around 80-85% during the race.
I understand (not a doctor though) that overexertion with RA can have a more significant effect on the body than in a non-RA person. I've got a couple of friends with RA, and I've seen them have a similar reaction to what you describe, even going so far as to faint after exercise.
I think it's important not to lose sight that RA is a systemic disease, affecting your whole body and interactions that occur within it, so things that don't affect others might affect you. It's really important, IMO, to listen to your body, and at the moment your body's telling you that something isn't right. I'd be tempted to take it down a notch, run a K, walk a K, drink plenty of water and electrolytes and see how you get on.