Hi, sorry you are having to go through so much. I too deal with RA and COPD among a couple other things, it is no fun at all.
Yes, RA does affect my COPD and reverse my COPD affects my RA. When either one flares up, you (your) body is under stress, causing blood pressure and/ or heart rate to rise, thus causing your heart to need more oxygenated blood, needing more oxygen then causes the breathing to enhance so to speak. Please keep in mind that RA not only effects our joints, but soft tissues, such as tendons, muscles, ligaments and yes our organs, including inflammation in the lungs.
Its a great idea to chat with your doctor about the 2 issues, maybe even try to get your Rheumatologist and Pulmonary Docs to work together and find a plan that will help with both. The trick is to get both under as much control as possible.
I so wish you much luck with all this, for handling more then one chronic illness is a tough job.
Best to you!