It took be several tries, but what worked was as follows:

Take a suitable length of coat hanger and loop it through the closed end of the spring. Then twist it several times. Make sure you leave a large loop (3/4")

Attached the hook end of the spring into the replacement retainer in the stem, fish coat hanger and spring through the hole in the base and orient the spring so that when the mirror is in the correct position (when mounted on the car) the loop end of the spring is perpendicular to the mounting post in the base and is on the side of the spring facing away from the base.

Disconnect the glass part of the mirror so it isn't damaged

Clamp the coat hanger firmly in a big vise so that the end of the base is is about 3/16" from the side of the jaws.

Slot a soft piece of wood (I used a paint paddle) so it fits over the coat hanger and to between the vice and the mirror base.

Adjust the vice/base distance as needed to allow you to lever the spring by maneuvering the base up down and to the side, against the wood. When the loop is over the post use a screw driver to push it down and seat it on the post.

Use side cutters to remove the coat hanger. There will be a remnant of it lodged between the spring and the post. I was tempted to let well enough alone, but risked busting off that white metal post by grabbing the remnant and twisting it with needle nose pliers until I got it out. Thankfully doing that didn't pop the spring back off.

The next problem was those crappy white metal mounting studs. I probably should have cut them off and drilled and tapped the base for new steel or brass studs. They had about one fastening left in them. Maybe next time.