Thank you for your question, Laura.  Fall is the best time to transplant established perennials, especially one that appears to be as crowded as is yours.  The timing is because the upcoming hot and dry months stress your plant's root system, in the same way removing it from the soil will do.  So, you are just compounding the potential problems.  (Note that this is just the opposite from planting bare root roses, which can be planted now.)

However, you can prune the plant can be pruned in March, and you can remove stems down to one-third of the remaining plant.  Just be sure to leave growth nodes (from which grow side stems and flowers), or you'll end up with a non-blooming plant.  

In anticipation of your eventual removal of this mature rose, you might find the following OSU article helpful, since planting details (except for timing) are similar for bare root and established plants:  https://extension.oregonstate.edu/catalog/em-9474-planting-bare-root-roses-spring

Good luck!