The great blog posts from Paul J. Drongowski (https://sandsoftwaresound.net/akai-mini-play-midi-mod-1/) explain in details how to mod the Akai MPK Mini Play, but only the older version. The MK3 version has slightly different internals, so I had to tinker a little bit for mine.
The picture says it all : 3 wires need to be soldered on the motherboard. The blue wire is MIDI out. The red wire is 3.3V, and the black one is ground.
Adding two 33 ohms resistors and a mini jack output gives TRS midi output, and it works very well!
Warning: the only dangerous thing to do is drilling the back plate for the mini-jack output. While drilling, I accidentally damaged the screen ribbon connector underneath. I was able to fix it with a lot of glue, but that was close...
p.s. : this is my first reddit post...