Right-of-way is never something that you can claim or seize for yourself... it has to be a gift, e.g., the other fellow has to give it to you, otherwise there's going to be a crash ;-)

Typically the rules on trails and multi-use paths are that hikers and bikers yield to equestrians (because of the horses' high proclivity to spook and bolt, sometimes taking their riders with them, and in any case presenting the danger to all others of a loose, out-of-control thousand pound scared animal); bikers yield to hikers.

In almost all MUP systems that I'm familiar with, pedestrians have right of way over cyclists, and your need to complete your interval doesn't overrule your requirement to safely pass any slower traffic and not to endanger anyone else on the trail.