You know, you could get a galvanized pipe cap for 5 inch diameter pipe and drill a hole out of the center to slide the threaded rod through. that would be way strong enough, and only $200 for one.

OK you guys are all washed up as far as understanding the problem. I may not have been clear or you may not have been reading closely. Here is a drawing of what the situation is. What I need is a piece of pipe that has a 3.75" OD. That will fit into the sleeve and just catch the exposed edge of the outer race.

The smaller bearing is not to scale, the 3.75" pipe would fit.

You didn't say you have a smaller access to the back side of the bearing. In that case post #13 is the most reasonable, short of your magic tool I've never seen.

a couple of not great photos of the devastation:

First picture ~ Piece of broken inner bearing race. Large bearing or small? Is it attached to something or just sitting on the tub?
Second picture ~ Chewed up face of what? Looking up/down the bore at what?
Third picture ~ Shaft with partial inner bearing race, into something with a threaded outside diameter.

Hm. Okay, I admit that when I watched the original video I giggled, because it reminded me of setting up a loom only not as complicated, because looms. I have nothing to add regarding the washing machine, because I strongly suspect that the designers planned for this. It may be more efficient to wash clothes by hand in future.

The bicycle tool in the first post would have to be spring steel to pass through the smaller bore then spring out to hug the bore and catch the 1/16th shoulder.
You could fake it using steel or even aluminum/copper tubing/conduit. Make the two cuts and use a nut/washer/threaded rod to expand and hold it spread after passing through the smaller bore.

It is spring steel. I discovered that a section of exhaust pipe at my local NAPA is the perfect dia to fit through the smaller end (2.5") I will make 4 saw cuts from the end of the pipe up a couple of inches and flare the resulting tabs to a bit larger than 2.75"

The tool is pushed backwards through the wider opening until the sprung ends pop into place. I'll see if I have the wherewithal to make a video.

The tool we used at the bike shop was a bit different from the one in the OP; it had stress relief holes drilled at the top of the saw kerf. I doubt (hope not) I'll need this tool again.

I took better clearer pictures last night and will try to post them.