I finally had some time last night to work on this again. I tried the rubber band trick, but it didn't work. Rubber bands that were skinny enough to fit in the gap between the hub and the freewheel were not strong enough to compress the pawls against their springs. So I tried dental floss. It was strong enough, but kept slipping into the little grove that held the springs, and since that groove had a smaller circumference than the outside of the hub, the floss was too tight to remove later, and wouldn't come out of the groove. I needed something a little wide, but not very thick, and it had to be strong. I pondered that for a little bit, and came up with ribbon for wrapping presents. It was perfect. That's the trick! Use ribbon!
So then I put a little grease in the bearing chase inside the freewheel. And put the ball bearings in, one by one, and pressed them into the grease so they would stay stuck in place for now.
If I was called handhandhand, I would have taken a picture of this step. But since I only have 2 hands, you will just have to use your imagination. I held the flywheel with the bearings up, like you see in the picture above, so gravity would continue to hold the bearings in place, and I took the wheel and flipped it over so the ribbon was dangling down. I brought the flywheel up to the hub so the ribbon went through the hole, and I just gently brought them together. Then I flipped it all over and took a picture.
I pulled the ribbon, and it came out without too much trouble. Just slow and steady tension while I wiggled the flywheel a little in place. I tried turning the flywheel, and it turned just fine. I could hear the pawls doing their job. Click, click. Click, click. Click, click. Then when I turned it the other direction, the pawls engaged the wheel and the whole wheel would turn. It was working perfectly. So I put ball bearings in the top, and put the ring back on.
The freewheel was actually pretty loose when I used both shim washers under the ring. It had been that way before I took it apart. So I tried removing both shim washers. Then it was way too tight. So I put one shim washer back in. Oh, and then I poured a little oil in there to really lubricate everything. It's just about perfect now.