Why do you think that it is the meniscus? Did you have any injury in this knee? How old are you? I would doubt that it is really the meniscus. Your intention of seeing a physio was good, to clarify what the problem is. Any chance of seeing an orthopedic/sportsmed doc?
How do you feel the lack of stability? I would target that with specific strength training like squats and stepups. Do them in front of a mirror, or with someone to correct you, and focus on good form. Also, balance training would be a good thing. Again with good form and a straight leg axis.
I wouldn’t prescribe a rest period. It looks like you can handle it well at the moment. As long as it is not getting worse, I would continue with training. But be very careful and honest with yourself. And ad the strength training. See how it progresses. If you are short on time better skip a run instead of some strength or balance training.
I hope that helps!?
Have a great weekend and a satisfying long run!