I found a solution, so I'm posting it here if you ever have to face it too.
I noticed that the snapshots created by Lefthand Hardware VSS provider didn't have any servers associated with it.
Even though I was advised by HP support that DP 8.1 supports LeftHand SAN IQ 10.0 and not higher I tried to install all versions of Lefthand VSS Provider including 10.5 and 11.0 on the backup server without any success.
It's not very well documented how this process should work, but by poking around with VSS I concluded that DP talks to VSS and then VSS talks to Lefthand VSS hardware provider to do the snapshot. I ran VSS debug which didn't reveal anything new.
However, after installing Lefthand VSS Provider version 11.0 I noticed there's a registry key:
LOCAL_MACHINE\SOFTWARE\HP\StoreVirtual\AASM\SupportLogsEnabled=0
Changing this to "1" and rebooting the server it enabled Hardware VSS Provider debug!
From the debug I could see that the snapshot is found, but cannot be mounted because it doesn't have any servers associated!
After that I installed the "unsupported" Hardware VSS Provider version 11.0 on the application server and new snapshots now have servers associated and now they could be mounted on the backup server.