I started on some similar testing of this issue on a 4-Slot to help out.
In my instance I have KOF '98 in slot 1, Metal Slug X in slot 2, League Bowling in slot 3 and MultiMVS in slot 4.
Upon turning on the cab, instead of starting in slot 1, it automatically goes to slot 3, pressing the select button, I can go to the MultiMVS in the next slot, choose a game then everything is fine the system reboots to slot 3, then the 4-Slot functions as normal with the attracts, coin, select button, etc. -- In the Neo Geo menu, the cart slot allocations are correct, and League Bowling is in fact Slot 3.
Changing the carts around in any other order causes the MultiMVS to prompt "file system not ready" at which point it also fails the SD Diagnostic test in the menu.
I have had no luck with the cart / motherboard saving any settings. The battery on the 4-Slot is just fine and it saves the settings of the other games, date/time, etc. I've tried multiple clearing of the BUFF file, clearing the NVRAM, power cycling, cleaning slots, cart edge, new SD cards, different cluster sizes, etc.
I have also found is that certain games will not load when using my 4-Slot: Crossed Swords 2, Twinkle Star Sprites and Top Hunter are the ones that spring to mind. They either get stuck in a constant state of green screen rebooting or glitch out to a solid screen. This seems to be an issue with 4-Slot only, popping the cart into a 1-Slot motherboard with no changes to anything, the same games just boot up automatically.