I believe I fixed it for me. TLDR: do something to keep the battery from moving up/down from the 3 inner connectors.
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Today, my controllers got to a terrible state where they were disconnecting all the time while playing a fighting game (quick hand movements).
As suggested on some other answers, I tried adding some folded paper between the cover and Meta's rechargeable battery, didn't work. But that got me wondering about those 3 inner plugs, and I started fiddling with it and noticed both + and - connectors on the controller are spring loaded.
That means the battery will move more towards the weaker spring. And it will keep moving as the springs' plasticity change their strength. While sliding the battery up and down to check, lo and behold! The controller would turn on, vibrate, then turn off.
In my case, it needed to move a bit towards the face of the controller, so I added some tiny, crumpled paper balls to the side of the bottom connectors and voila! No more disconnecting! I believe, as the top spring settles down with time, I'll need to readjust, perhaps even remove the paper balls. But for their current balance, this keeps the battery in place.
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Maybe the 3 pins are not essential to keep it on, but perhaps it causes an issue if the battery moves vertically enough for them to touch the wrong connectors on the controller and cause a weak short circuit, or some unexpected miscommunication that crashes the controller firmware.
