I'm not sure that cutting it back a bit now 'would' stress it, but waiting until November is a fine alternative.
as to replanting, I had a very old climbing rose planted right next to me front door. It had to be removed because we were having a large new porch built. I took loads of cuttings, cut it right back and dug up the root. I replanted the root in a completely different place just in case it would survive. Not only did the rooted bit regrow but all the cuttings took!
I can't remember what time of year I did this (it was about 20 years ago) but probably it was spring or early summer. It would be worth a try if you really wanted to move it.
Your rose might not be a climber, many bush roses will grow very tall if left to their own devices (I know this because I have a few, there's a 6foot+ one waving at me as I type
though I have cut it back several times over the years...)