So, I gave up for now...
For the record, here's what I did:
I didn't get one of the MDR connectors mentioned above and I choose to completely desolder the MDR connector from the MPC HD mount. A bit difficult but no problem so far.
I got one of these SD card to IDE 2.5 adaptor and a 2.5 44pin ribbon connector that I cut in two. One side connected to the SD reader, the other side directly soldered to the PCB.
Then I soldered each connection one by one following the schematic. A real nightmare and very prone to error, cold joints etc, I won't recommend anyone doing the same thing, these flat ribbon connectors are not made for that but it was worth trying.
The SD card reader is powered on, but the MPC won't format the sd card, there's something wrong and honestly it's too difficult to debug with so many connections. I had issues before with my MPC not loading the files properly, firmware or hardware, it's hard to say, so it could be this or simply one of the little cables not properly soldered. I won't try harder right now, I desoldered the 44pin flat ribbon (it's too fragile anyway) and will re-solder the MDR connector.