My concern with either a shoulder or perch tether is that the bird would fall in a direction where you wouldn't be able to get to it quickly, and on a bike (like in a car) if you go grabbing for a panicked, flapping, bird you cant reach, or worse, can't see
either, you're at high risk for an accident which could injure both of you. If the bird is on a perch and falls forward, you're not going to reach it easily to help it right itself. If it's on your shoulder and falls back, then you can't see it either. One scenario has you pulling on the leash trying to right a wildly flapping mass of yellow and green feathers while that mass is digging talons into your neck and a beak into your earlobe. At the same time you're trying to stop on the one big downhill in the middle of summer where you get to have some speed and coast so you can cool off. Of course, it happens that every person on the block just watered their yard and the street is wet ... and, well, I could go on, but you get the idea.
My thought is that if you're going to use a harness of any kind, it needs to be at least a two-point harness. Preferably a three-point. If you're not familiar with the terms, it means that the harness is secured to the base (person or perch, whichever) in two or three places. Ideally they would be front, back, and side, and short enough that even if the bird slipped and fell back, the tether in front would stop the fall, and vice-versa.
I don't know exactly how this could be accomplished - possibly with a regular flight harness and either three tethers or a three-way adaptor you rig-up somehow.
Of course, all that thought and preparation is useless if you happen to have a good old-fashioned bike accident. During my stint in the ED, I saw more than a few victims of bicycle accidents. People can get pretty banged up, and in situations like that, we're far more sturdy than a birdie.
So, those are my thoughts. Take it for what it's worth, but if it were me, I'd take my birds on a walk!