lovesblooms .........
Thank you for providing the background of your roses.
I agree that your climbers have not been in the ground long enough to grow the root mass they would want before they would put on significant top growth and bloom more prolifically and giving them another year is probably very wise, especially since you say they are healthy plants where they are currently located. That third year really does make a difference. I have found that to be true for most of the roses I have grown even when they are not climbers.
In my current garden, I don't think any of my roses reach maturity until their 3rd or 4th year in the ground and then they seem to be healthier and bloom more prolifically. Of course, as usual, it depends on the rose.
The rose growing in the shadier location may have additional problems other than light deprivation which is causing it to look like it is just surviving. Is the additional shade due to the growth of near by trees ? If this is true, the rose may be impacted by root competition from the trees and you may decide to relocate that plant where it does not have that issue. If the shade is caused by a structure blocking the light, that's a different issue. You can use this third year to observe whether or not the plant is going to take off, but do give it the third year. As
@Xeramtheum posted above, some roses can grow well in spite of the root competition from trees. Ralph Moore, a world famous, and world class, rose breeder used to say, "If there is a way, the rose will find it." Xeramtheum's post certainly confirms this statement.
As for suggesting specific cultivars for a weeping rose, I am more comfortable describing the plant habit you want to look for because our climates are so very different, but will suggest three that I know have the plant characteristics that make for a good weeping rose.
You are looking for a rose that is low growing and spreading with limber canes, so that the weight of the blooms cause the canes to hang down. Many people plant almost any miniature rose in a hanging basket thinking it will hang down, but most of these roses really do have an upright habit and never weep. Often, the type of rose you will be looking for is described as a ground cover rose, or is used to make a standard (tree) rose. You can Google "weeping rose" in images and find several good candidates. (There are some incredible photos there.)
I am also hoping others on ATP may give you some specific suggestions.
Of the three I am going to suggest, I am only growing one right now and that is 'Green Ice'. I am growing the rose in a container, but it has the perfect habit for a weeping rose as you can see in the photo below. The other two that I have seen growing in hanging baskets that kind of blew me away were 'Sweet Chariot' and 'Red Cascade'.
These roses can become larger plants than you may want, but don't mind being pruned back to a smaller size.
Good luck with your project and it would be wonderful if you share photos as the project comes together.
Smiles,
Lyn