Maybe your seat height is an issue. Try varying the position of your foot on the pedal - changing the contact point from instep to toe. This effectively allows experiment equivalent to quite a large extension of the seat post which you can try on the fly without adjustment.

I have occasional trouble with my left knee and find that I can get relief by this method. It seems a lot less painful (when it happens) if I push the pedal with my toe rather than the ball of the foot or instep. This trouble always starts for me when I have been hill climbing in too high a gear.

The last point brings me to this question: Is your Brompton over geared for you? You mention your mountain bike not giving you the problem. Maybe that is because it runs lower gears, or you have the option of them when the knee starts to give trouble. You can lower the gear ratio of the Brompton by getting a smaller front chain wheel. Pushing too high gears puts a lot of strain on the knee, especially if the seat height is too low.