The term "BX" is sometime used a little loosely but I'm going to presume you have some kind of metal jacketed cable without a ground wire. For those kind of cables, the jacket is the equipment ground. With type AC cable there is also an aluminum bonding strap but that can't enter the box.

You need to either replace the box with a metal box or replace the cable with something with a "real" ground (NM-B [Romex] or metal clad cable, type MC).

I'm not sure an old work box is designed to support a fixture. Maybe someone could comment on that.

Yes, I'm talking about the metal jacketed cable without a distinct ground wire. I was hoping to not replace the box, since the hole is round, I'd be patching drywall, etc. and the room just got painted. I was hoping to be able to attach a wire to the jacket and extend it into the box. No solution for that without swapping the box? BTW, the walls have metal and there is one adjacent to the box, for what it's worth.

Plastic boxes are not listed for use with AC.

Without a cable clamp there is no way to obtain contact with the armor.

I picked up a grounding wire clamp, size 3/8" to 1". Menat for pipe, of course, but think that will do it?

There's no way that will meet code to accomplish what you want to do without replacing the box or cable. Plastic boxes are simply not approved for use with that kind of cable. Can you jury rig something? Of course, but it's not recommended.

That might provide an electrical path to ground, but it won't be code-compliant. Please refer to Brickeyee's comment.

Got it. Thanks much.

If it were MC cable (with a ground in it) would the plascit box be ok? How would the cable be connected to the box?

Fan box.

It's more than you need, but it will solve your problem.

Closest thing I can think of to what it is that you really need: a metal old-work box.

Here's an example

"Fan box."

That would do the trick, and can be installed through the existing hole in the ceiling.

The clamps in the box are most likely for NM cable though, so you need to use a knockout with an AC cable clamp.

Hint: Attach the cable clamp to the cable and thread the wires through the knockout hole.
Place the box in the hole and pull the cable clamp into the hole and put the lock nut on and tighten.
Fasten the box to the bar.