If they were suckers, they would usually be of a species which has smaller leaflets and more pairs of leaflets than the ones in the picture. And there is no difference between the leaflets on the flowering stem and on the new shoots. So, you seem to be right that they're not suckers.

You shouldn't have a problem recovering this rose - I've had them come back from less. The thickness of the new shoot is an indication of a reasonably healthy root system.

I would prune out the dead branch, and the few dead twiglets on the new branches. Others will tell you whether it's possible to stimulate growth in this case by pruning, and how to do it. Bearing in mind its sentimental value, I'd be inclined to leave it in position to gain growth, then take cuttings and start a new plant for the new position, and only move or get rid of this one once you have a plant secure in the new position.