Herpes gladitorium can be transferred either from a mat or directly from someone who has a cold sore and comes into direct contact with a wrestling partner. It is not common as you may know.
Are you saying that the person gets cold sores on their lip? Is that the concerning area?
I have zero concern about this person touching their face, assuming that is where the outbreaks are occurring, and then touching surfaces in your home as a method of transmission.