|I was thinking of planting a New Dawn Climbing Rose in combination with a pyracantha that is|
|Both of these plants have large root systems and heavy top growth. They both need full sun and good air circulation to stay healthy and grow well. The pyracantha could be trained as an espalier against a wall, and a New Dawn climber could also be trained to a wall, but I think both together would be too crowded. Also, you will need to have easy access to them for routine pruning and maintenance.
Usually if a rose is grown in combination with another plant it is grown up a tree with a bare trunk, this allows access to the rose and the two don't grow tangled together. Or, there is a clematis vine trained onto the rose support, typically a variety that is cut down each spring. Again, this allows you an opportunity to prune and train the rose easily.
Depending on the exposure, another thing to keep in mind is you may find you need to take the rose down and bury the canes each winter to protect it.
All in all, I don't really think this would work out very well. On the other hand, you can always try it and see how it works out for you.