Here's the solution (Thanks to deajan)
Cobian's service is launched as localsystem or as a system service. When using Cobian as system service, it actually acts under a user account which is granted network service activity.
The thing is that VSS operations are performed as NT AUTHORITY\Network Service which is not granted COM access.
You can get this granted by launching dcomcnfg.exe, browsing to to Component Services, Computers, and right clicking on your Computer. Get to the COM security tab, and grant Network Service (or SERVICE RÉSEAU in french) local access.
This should get rid of your Acces denied messages by VSS.
Additionnaly, you can grant Network Service full control over HKLM\SYSTEM\CurrentControlSet\Services\VSS\Diag
You can also check this key HKLM\SYSTEM\CurrentControlSet\Services\VSS\VssAcce ssControl which should at least contain the DWORD "NT Authority\NetworkService" set to value 1.
You may also add a new DWORD like "DOMAIN\MyCobianServiceUserAccount" and set its value to 1.