Restore database from individual datafile backups
Jay, this isn't officially documented, so I'm speaking only from my own experience.

In RMAN, the command BACKUP DATABASE ... is synonymous with BACKUP DATAFILE 1, 2, ..., n-th ....

Also the command RESTORE DATABASE ...  is synonymous with  RESTORE DATAFILE 1, 2, ..., n-th ....

I'm not sure about RECOVER DATABASE; it might be also a synonym of RECOVER DATAFILE 1,2, ... n-th , although  I've never tested the latter. In your scenario, the former would work as expected.

In particular, neither RESTORE DATABASE or RECOVER DATABASE require all the datafiles to come from "a single backup of database" (from a single run of BACKUP DATABASE). Actually, RMAN doesn't even have a concept of "a single backup of database".