DS mode launching GBA mode is possible but you don't retain any memory even if you could somehow force a jump to it to act as a cheat engine. To that end nothing like the GBA equivalent of
http://www.chishm.com/NitroHax/index.html if that is where you were heading.
Your options then become dumping the saves which can be tricky with those multi carts -- GBA carts top out at 32 megabytes which the first 3 games would take up, maybe one more if the trimming works in your favour, and thus something fun will need to be done to handle things. I don't know what they would have done here either -- most pirate multi carts for the GBA are either repurposed flash carts (doubtful here if it is all the games and revisions plus a menu) or rather more basic affairs (several takes seen over the years,
https://gbatemp.net/threads/sram-bank-switching-on-bootleg-cartridges.547662/ being a twist beyond standard protocols). 12 games even without 4.5's 512K Flash would also put it above the 2Mbit that baseline cartridges top out at so not sure what is being used to pull that off.
Said trickiness is also why your AR does not work either, same as
https://doc.kodewerx.org/hacking_gba.html#nonstandard .
If it is a repurposed flash cart and you can figure out how to write it then you might be able to hardpatch the cheat in with something like
https://gbatemp.net/threads/gbaatm-rebirth-gba-auto-trainer-maker-a-new-hope.564321/ but that is getting a bit radical at that point.
I have not looked up what the softlock is here either (I knew the one for 4 on the DS
https://www.nintendo.com/consumer/systems/gameboy/trouble_specificgame.jsp but these I am less familiar with) but depending upon the nature of the softlock if reverting to an earlier save might cure it/allow a dodge then you might be able to force that if the game uses alternating save banks by turning off during saving but that risks everything.
If you or someone you know has one of those nice analogue pocket devices then you could actually snatch a savestate from that, edit the savestate (AR codes are just instructions on editing memory addresses, which you have the complete ones for by dint of having the codes) and restore that (I did something similar in
https://gbatemp.net/review/analogue-pocket-gb-gbc-and-gba-handheld-fpga-based-player.2081/ ). It would probably have to be a device of that potency as well -- the more classical dumper-emulator combo that most historical clones are will probably only get the menu you use to select the games.
I don't know what goes for hacked GB players these days but I could see a path there
https://github.com/FIX94/gba-link-cable-dumper/releases for a jumping off point, though it would take more work and basically be a custom dumper.
Re photos. If you can post one of the PCB (even on an external site and break up the url a la www . somesite . tld / blahnumbers , or say whatever image host and the unique bits so we can reconstruct it) I and probably several others would be interested; what you describe is a step above what we normally see for bootleg/pirate/repro carts, and indeed several vintage flash carts.