You can't have a 7.7.3 master with an 8.1.2 client. I'm not sure whether you're asking that.
I expect an 8.1.2 master to work with a 7.7.3 client, with one caveat. We had a mix-up in 7.7.3 in the format of the Discovery data (NetBackup Discovery Framework service - nbdisco). This caused GRT backups to fail for Exchange GRT and SQL Server if 7.7.3 was mixed with any other version. You can get an EEB for the 7.7.3 client to make it compatible with newer master servers. Reference ET 3894489.
The EEB originally was for a 7.7.3 master with a 7.7.2 client, but it also applies for a 7.7.3 client with a later master. The EEB makes the NetBackup 7.7.3 installation compatible with other versions before and since.
After applying the EEB to your 7.7.3 client, restart nbdisco and wait for new files for that client to appear in NetBackup\db\discovery before trying your backup. (Note, the EEB makes that 7.7.3 client incompatible with an unpatched 7.7.3 master, but if you've upgraded your master server to 8.1.2 that should not apply.)