How long ago was your last in-person visit to the walk-in clinic? My primary, who is very reasonable, won't call in refills unless he sees me face to face at least once every six months. The local walk-in clinic insists on seeing you every 3 months, otherwise you're not considered a current patient. That's fairly normal practice.

Seeing that it's your blood pressure meds and you can't stop taking it because you're right about the risk of arrhythmia, maybe you'll have to bite the bullet and go in and let the clinic get the insurance payment because there's nothing more important than your life. Ask the doc if s/he'll give you five refills on the new prescription. That way, you can call the pharmacy when you need a refill and they won't need the doc's okay each time. That'll take you out six months.