There are question that are extremely difficult to answer because I always recommend that homeowners receive on-site technical assistance regarding these sort of questions.
I do not believe you need to cut down the tree to solve a "root" issue; however, I'm not going to say that it is "safe" to cut the root. Any sort of root excavation can potentially have deleterious effects on the tree; though the impact can be reduced if you conduct the root pruning through a professional tree service company.
You may want to try to contact your local University Extension office to see if they might be able to send someone out to your property for a quick evaluation.
Best of luck!