Some issues. I don't have your email address, and private messages don't appear to allow attachments. Nevertheless, this might be useful to other SA22 owners, so I'll just post it as a reply -
(The numbers are from the Browning 22 Semi-Auto Parts List)
There are more pieces involved, and you may be missing one, unless it is up inside the Barrel Lock
The pieces involved are
1 Barrel Adjusting Ring Follower Spring
2 Barrel Adjusting Ring Follower
3 Barrel Adjusting Ring
4 Barrel Lock Ring
5 Barrel Lock Spring Plunger
6 Barrel Lock Spring
7 Barrel Lock
The first photo shows these laid out in their respective positions. Note that 2 is smaller than 5 and has a screwdriver like tip so that it fits the grooves in 3 and "ratchets", while 5 just acts like a detent so 7 "clicks".
The second photo shows 5 and 6 assembled in 4. The next thing to do is put 1 and 2 in 7 and slide it into position on 4. The tricky part is that the long axis of 2 has to engage the grooves of 3 as you do this. (If it goes on sideways, it will assemble but won't click. Slide it out and try again.)
I've found that you can get 1 & 2 started by lifting with a small screwdriver or dental pick, then press in on 5 while you push 7 into its assembled position. The third photo shows what it looks like assembled.
From your picture, I think part 5 shot across the room somewhere when you slid 7 back...the barrel lock won't stay in position without the detent piece.
When you get a replacement, I suggest working under a sheet or inside a garbage bag to try to contain part 5. It's really hard to find in deep pile carpet...