Instead of going to a surgeon, perhaps you should see someone who's an expert on bike fit and related injuries.
I get slight knee twinges (inside the joint, not the tendons) when riding the Mini for a while. I'm pretty much convinced it's Q factor.
But in your case since you don't use feet position thingies on the Brommie it could be a few foot position issues. I would try it out with some clipless pedals just at least to see the effect. Foot position is really important.