So it looks like I found out some more information from other Musicomlab users.
Apparently, to toggle the H9 tuner, you need to send MIDI CC with value of 64 and above.
When you engage the global preset with FS1 on and the CC100 attached to it, the value is 127 (on normal mode).
When you git one of the presets again (P1-P4) the ML send CC100 with
value of 0 (since you turned off FS1) which is ignored by the H9.
Then, only when you turn F1 again, it send CC100 with 127 value which toggles the tuner off.
Not sure I'm going to be able to setup the tuner with this controller and the H9 how I described and wanted. I'll keep trying other solutions in the meantime.
Also – Is it possible to have an empty preset that can engage the tuner? If I could access the tuner via a PC message that would certainly be a lot easier!