[22GT]

The lever inserts straight down into the housing. You should have to press the buttons in just to get it into place.

Do you have the special very thick hardened washers for the shoulder bolts that hold the handle on?

When you shift into reverse, the large cylindrical pin in the shifter protrudes from the right side, activating the reverse lights. If that does not line up, you have the switch on wrong.

 

[22GT]

This is what I have. When I took it out, this is what was on the right side. If I put the 7B125 spring/cup on the right side, it'll just fall out. There's too much room in the housing.
There should be a spring and cup on BOTH sides. A tight fit, then.

You can see the washers in the pic.
They appear to be correct.

When you say "the large cylindrical pin in the shifter" you don't mean the shifter handle (below), do you? From your pic, the switch button seems to line up with the cylinder that everything rotates on.
No, not the handle. This: