using C clamp on ceiling beam to avoid screw holes - will it hold?
I don't think your plan to make a custom bracket will work for a number of reasons.  It's way larger and heavier than necessary, it will be very hard to make a spring clamp of that size to make a pressure clamp, and it will be hard or impossible to slide it on over VHB tape.  VHB tape may or may not hold up to the heat generated by the light.  

It's always good to be careful about drilling holes in a beam, but the tiny holes needed for a few small short screws to hold up LED panels won't have any significant effect on the strength of the beam.  

If you'd still rather not drill the beam, you could hang the lights from the joists near the beams.  If you want a surface mount, you could mount them on the short side, that way they will shine on more of the room.  

If you don't mind hanging the lights from a v-shaped wire or chain, you could hang them from eye screws into the joists on the longer side of the ceiling.  That will keep the lights level and get them closer to the center of the room, which will give you more even lighting.  It will also let some air circulate over the light, which will help the heat dissipate, which will be better for the LED fixture and for the ceiling.  

If you are really set on hanging them in that exact position, you could make a small bracket to go across the bottom of the beam (perpendicular) and attach to the ceiling on either side, either with toggles into the drywall or preferably with screws into the joists.  Not something out of 2" tube steel, more like 3/16" flat stock.