Ok So I tried something else.
I uninstalled everything.
I reinstalled the game (original version from 1998 + official patch 1.02)
I reinstalled the latest version of Aali's OpenGL driver (v0.7.11b)
Then I installed FF7Music using Covarr's latest installer.
At this point I tried the game with the RemasteredOST (FinalFanTim's) profile from FF7Music and everything worked just fine.
Then here is what I did in order to be able to use Anxious Heart:
I copied the Anxious Heart folder containing all the musics from my previous installation (which I previously backed up) to my new one (in FF7Music folder).
I recreated Anxious Heart's FF7Music profile in FF7Music.ini by copying from my previous installation backup.
And Everything is OK!
So it is possible to use Anxious Heart with the French version of the game.
I was just wondering why does the EXE from Anxious Heart manipulate the ff7.exe?
What does it do to it?
Is there something that will be missing from my installation the way I did it?
PS: I think maybe I should ask these questions in Anxious Heart's thread.