Preferably i would say that lets keep the SG name with DR as well for no further problems ...
and then when the active server goes down, just rename it to the original active node db name:
StoreA-F (Storage Group on Passive Node in Recovery Mode)
MailboxesA-F.edb (Mailbox database on Passive Node in Recovery Mode)
Yes you can do that as well ........
Just try with Database portability with a test SG and DB and see what happens.... that would give you a better and clear picture ...