We haven't seen the Klettersteig, and we were there last summer.
But we have experienced the Dorffest---by pure luck we arrived in Mürren on the right Saturday a few years ago. It must take place on the first weekend after Swiss National Day (August 1). At the train station, there were people passing around trays of small wine glasses and cheese samples, all free. We had to get to our lodging (Chalet Fontana), down the main pedestrian street. People had already assembled for the parade, and we were greeted by waves and cheers as we (7 of us) marched along with our roller bags. We checked in and found places at the windows to watch the parade from right there. After that people went to the Sports Center for food (sausages, beer, etc.). The cows that participated in the parade, all decorated, were tied in a grassy spot near the Sports Center. Young men that paraded with cowbells sat at the Jägerstübli and sang/yodeled for a couple of hours, then marched with their cowbells to another venue down the street. And then another, and another. This went on well into the wee hours of the morning. It was a great day.