Hammerli 208 fix advise needed

My Hammerli 208 International started to develop a Failure to Feed problems; unfortunately the problem showed up in a Timed fire portion of the match which cost me 40 points and a move to the Expert classification due to disallowed alibis.
The first round goes downrange, but the second doesn’t feed (on and off) into the chamber but stays  in the mag. The same problem occurs with my second mag as well. The issue has shown up before, on and off, but doesn’t occur every time, which makes the gun unreliable.
I clean the gun after about 500-600 rounds and completely disassemble the mags after about 2500 rounds, but clean the followers and mag lips and whatever I can reach with a Q-tip into the mag when I clean the gun.
I noticed that the Slide catch is very wobbly, not just up and down, but side to side, when I take the grips off. I I try to tighten it a bit, it stops moving freely, and doesn’t function well, so I loosen it again. I don’t know if that causes the problem or not, but the slide does stay open on the empty mag every time.
Can anyone offer any suggestions for the fix?
Also, can anyone offer any suggestions, now that Larry Carter is gone, who can service these pistols?
Thank you for your input.

Although I don't believe the slide stop is the problem, it should be addressed.  The slide stop nut should be a bushing with a nut head, that rides inside the slide stop.  The screw and nut should be tightened down snug and the slide stop should ride on the bushing part of the nut freely.  Also, one end of the spring should have a 90 degree bend that sits in a respective hole in the frame.  There should be pressure from the spring holding the stop downward, not just gravity.

Now, to the mags:

If these are Hammerli magazines of more recent manufacture, I suspect they are your troubles.  There are numerous wear points on them, one of which is chatter marks in the cross pin slots. This can be seen by looking at the leading edge of the follower slots.  This edge should be smooth along its entire length. If it has "serrations" they must be removed.  Other, less likely troublesome points, are the inside edge of the extreme rear of the rear lips, where the fully loaded top round is held, and the front tip of the left rear lip which gets a burr from peening of ejected rounds.  As for the inside rear of the rear lips, there is often a rim indentation that shouldn't be there, and sometimes there's a crack on the top rear of the mag at the 90 degree cutout where the lips rise from the back.

Edit addition:  I read that you do clean your mags, but you might take a closer look at the inside front area to make sure there isn't a buildup of debris that may slow down the raising of the top round.  Sometimes, this area gets a bit sticky.

Ed, thank you so much for the extensive explanation. I will carefully examine everything you eluded to.