I'd be careful about severe pruning. I killed mine by trying that. I think Dublin Bay is a classic example of how some climbers grow. As they get taller they start to drop branching and leaves towards the bottom so they can send up what's needed to the top. Thus you get the "bare knees" look on them.

Yours doesn't actually look as leggy as mine did when I cut it back so I would just let it be. I would say that as new canes grow out start training them back toward the trellis right away. Don't let them get hardened off before you try to move them to the trellis because DB's canes get very stiff as they mature. I snapped a large one off of mine because I waited too long to start training it back to the trellis. At this point you might be better off just using twine to hold those canes up against the strong winds but not try to move them back to the trellis.

If you want to slowly improve the shape over time I would suggest that you cut off only 1/3 of the oldest wood on the plant. That means if you have 3 old woody canes just take one of them out this year. Then wait and see if it gives you some new basal growth next spring. Keep taking out the oldest wood one or two canes at a time each season until the plant rejuvenates itself and you can move the newer canes back to the trellis while they are young and more flexible.

Thank you Seil for your experienced advice, I appreciate it.

The rose looks good to me.

If you want climbers to be their fullest, you need to train the shoots to go horizontally, not vertically. It's a pattern called espalier. Not sure how you would do that with your setup. You need some more lattice off to the sides. What happens is the new growth will go up vertically from the thick horizontal branches and those will produce blooms. You can then train the new growth on the ends to extend even further horizontally if you want. Trim off the old shoots when they stop growing this year.