That's happened to some of the
roses that were here already when we bought the house. I would sure transplant it instead of just cutting it off. If you just remove it like a sucker and clip it off all the way to the ground, it's still gonna come back there anyway. So, you have nothing to lose by transplanting it!
I've just taken the shovel and cut down between the original bush and the sucker, then scooped it out, keeping as much dirt as possible around it, and had another hole waiting to set it into. Most of the time when I do this I end up with a bush that grows red
roses, but a few times gotten pink ones.
Cindy