Definitely get rid of the ivy. Sometime in the past, people in my part of town decided that English ivy would be a good way to control erosion. Maybe so, but they never considered how invasive it would be. Every year of two I pay someone to strip it off the brick on my house. and hack it back a yard or two. It appears to be impervious to Roundup.
Unfortunately the ivy has taken over many of the trees in the neighborhood. Many younger trees have succumbed and majestic oaks are suffering. I have tried peeling the ivy off the trunks, but the ivy has developed into vines that are several inches thick and almost cemented onto the trunks. I fear that many of these beautiful trees are doomed.