You can fish the wire over the beam. I'm not sure what NEC requirement you're talking about.

I'm talking about the 1-1/4" bore hole clearance from a framing member surface to help prevent nails and screws from penetrating the wire. The floor above the beam is almost 1-1/2" of wood, but does this satisfy the no conduit or no cover plate needed distance?

Is 1-1/2 larger than 1-1/4?

The actual clearance is from the face of the member and ONLY applies to cables running in bored holes, it is NOT a general clearance requirement.

If you have a gap between the top of the new beam you can run a cable in through the gap as long as you can fit the cable into the gap without tearing up the outside jacket.

A loose cable tends to move if hit by a nail or screw, unlike one in a tight bored hole that cannot move.

Thanks for the clarification Brickeyee.