I always advise cutting back severely a transplanted rose, even one suposedly dormant. No matter how big of a root ball you get, you still leave most of the roots behind. The larger roots which you see are anchor roots. The tiny feeder roots which do all the work of supplying moisture and nutrients easily fall off when a bush is moved. These must regrow before the bush can absorb moisture. Leaving a lot of top growth severely stresses the bush as it cannot supply the needed moisture.
In your case, as you can see, you'll have to prune anyway as that portion of cane that drys out will die.
Prune back now . Perhaps it's not too late to save it.

How much should I prune off? With some canes, I can pretty much tell by looking that they're dried out. But with the larger (over an inch diameter) canes - I have no idea. Should I just proceed with cutting off as many of the smaller canes as I can and then cut off anything that's clearly dead as the other buds (I hope) start to swell? I have other roses, but this is my only climber, and frankly, I'm not so clear on pruning it, anyway!

Let the rose decide what to keep and what to junk. It will have a better idea of what is salvagable than you will. If you are sure something is dead, it can go, but I'd be pretty sure about that before cutting.

We've learned that not pruning back as part of the move, and waiting to see how much top the rose can support works pretty well. Sometimes you end up with a surprisingly big bush when it's all done. Sometimes you end up with what you would have pruned off anyway. If the rose is defending its turf tooth and nail, we definitely get rid of the obnoxious parts, but aside from that...

Moving can very much upset the timing of plants. This is a new plant, at least in that spot, and those can leaf out mysteriously late or mysteriously early.

You have contradictory advice from two genuine experts. Sometimes we don't know enough to be sure what to do, so you will just have to make your own decision.

I haven't moved enough big rose plants to be able to speak from experience as to whether you need to prune or not. I can tell you this, though: a dormant plant with no leaves does not lose water at any great rate, so I doubt the rose is dead.

The shrivelled canes are likely dying or dead. Green inner bark is the sign of life in a cane, so check for that. Removing dead or dying canes won't reduce the water stress much, since (if the inner bark is not green) the rose has already shut them down. However, removing them might reduce the chances of botrytis canker, which often starts in dead or dying bark.

I moved E. Veyrat Hermanos, which was turning into a giant of a rose after a year in the ground. I had already cut it back once in its original place, and it grew back vigorously. Before moving it I cut it back to about two and a half feet. After transplanting, it adamantly refused to grow so much as an inch and about eight months later is exactly the same size. It now is showing some signs of wanting to leaf out but I feel this plant has been greatly set back by either the pruning or the move, or perhaps both. In general I've noticed that roses I've moved lose at least a year of growth, and I've become much more reluctant to move roses unless absolutely necessary.

I moved Jen's Munk today. It had grown to 6 feet tall and was shading some roses behind it. It had also gotten quite big around and was growing out into the yard and interferring with lawn mowing. It would bite me as I drove past on my garden tractor.
Anyway, I pruned it back to about 12 inches, dug it up, cut two rooted canes off it and potted those to sell at our annual society plant sale, and planted the larger portion in a different place. I didn't hesitate cutting it back because I know it's better for the plant than trying to salvage the whole bush. It was still dormant with a few buds beginning to swell. It may not reach 6 feet this year but it'll be fine next season.
And I won't have to worry about checking on it or pruning off anymore dieback. When I checked it before cutting it back, it appeared to be green to the tips.

Is it tougher to move a plant in a climate that has no true winters? Perhaps the plants get too big--the plants can't stand the huge reduction in size once people get around to moving them. Or does the winter reset the biological clock of a rose?

A large shrub or climbing root will have a much larger root system than a Hybrid Tea, even though both are on Dr. Huey. I just dug up a Tropicana that was in the ground for perhaps 5 years. The anchor roots were pretty thick, but probably not a whole lot longer than when I first planted them. On the other hand, a big shrub can grow roots several feet long. Transplanted bushes often grow better once I've eliminated the tree root competition. I also carried off a 5 gallon pail of invasive tree roots to the compost pile.

I typically move as much of the plant as I can. Even though I'm I stroke survivor who is just 5' 2", I can move a pretty big bush.

Thanks for all the advice! When I checked it last night I did see one or two growth buds that looked like they were swelling to leaf out. So I think I will wait a tad longer and see what happens. After the next few days, I may have more buds swelling and some canes that definitely look dead that I can cut off. I'm pretty sure there are some of those. If it were a normal HT or shrub, I would have no hesitation about pruning it back severely, but as a climber - well, I've been waiting for 3 years now to get decent bloom from it and finally said, if it's growing 15 foot canes and not blooming well, I have to think it's the amount of light in that spot! I was just hoping to be able to get decent bloom this year, but it may be next before I see any improvement.

You should expect to cut an established climber down to 18" - 24" when moving it. You can also anticipate getting very few, if any, blooms that first season. This is especially true if its a Spring bloomer.