You’re right, the problem is the Carrier Latch, and a new one will just drop in.
The catch is that prior to 1958, the part was screwed in place. If there’s corrosion or you strip the screw, you’re screwed. In 1958 Browning drilled the hole all the way through and started using a roll pin to secure the part. Now if there’s trouble you can just drive it out. Care must be taken when installing the new latch that you get the holes lined up exactly because it’s easy to damage the new part.
I had this problem on my 1965 Light Twelve. I tried to fix it myself by bending the front of the latch so it more fully blocked the magazine. It worked for about 50 rounds. So I took the gun to the famous A5 gunsmith, Art Isaacson. He told be to bend the rear of the latch to make the repair. But he couldn’t get it to work after my own attempts, so he replaced the part. No gunsmithing required.
One other thing, a gun with this problem will operate normally when loaded with only two shells. The next-to-load shell is restrained by the Locking Block Latch pinned to the bottom of the bolt, bypassing the Carrier Latch.
Here’s Art fixing my gun: