I played the Windows version for a while, and just installed the Linux version.
Is there a way to recover my saves ?
Cloud sync does nothing, and I tried to copy the save files from Windows in a .config/unity3d/Harebrained Schemes/BATTLETECH/CO/SGS1 that I had to create (.config/unity3d/Harebrained Schemes/BATTLETECH/ was axisting but not the C0 and SGS1 folders) , but it didn't work.

It's solved now... And I don't know exactly how.
What I did is: deleting the folders I made (C0 and SGS1, the later one being in the first one) , launch a new campaign, do a save, and transfer the Windows saves files to that newly created save folder ( ~/.config/unity3d/Harebrained Schemes/BATTLETECH/C0/SGS1 ).
The only explanation I see is that I made a mistake in the name of C0 or SGS1 ?? Perhaps...