Possibly if it is the type that should be pruned after blooming - Type I. You really need to move it after it is dormant.

Well I planted it in front of my climbing rose bush last year, hoping it would fill in what the rose doesnt. I thought it died then. It came up this year, but the rose is growing more, and bushier, and is covering the clematis, the vines are creeping across the ground, away from the rose and trellis, and the leaves are all dying at the base. So I guess I'm wondering if I should leave it be for now and move it later, or move it now before the rose completely covers it and I have a repeat of last year.

Depends not only on the type of clematis but how well a job you do digging up the rootball and how long your growing season is. Clematis often sulk, depending on the type, after transplanting and spend more energy on root growth and getting settled in. In fact, that is what you should be encouraging when you transplant any plant. Let the plant get settled in and don't expect the blooms. If blooms are important to you, leave the plant where it is now and move it either late summer/early fall or next spring when growth commences.

Many people grow their clematis with roses. Any reason you can't have the clematis climb up onto and into the rose plant itself? Just make sure both plants get fertilizer and watered well over the course of the summer, especially if Mother Nature doesn't cooperate with the rain.

As for the leaves at the base of the clematis turning yellow and falling off, that is typical for some clematis. It is called bare legs and isn't a big thing for the plant. It may be more of an issue for you but the rose foliage can camo it!