Hello Tim,
Case backups should go through an upgrade process when restored to get them to the current version of schema on an appliance, whereas the full application upgrade has to go through these stages when upgrading - it is a bit different for a case.
I've never tested with 6.1 direct to 8.1 in a case restore, it doesn't state it isn't possible - the only restrictions are the ones you've mentioned with regards to application upgrades, I can't guarantee with 6.1 -> 8.1 as I've never tested it but if you are able to provide me with the stock 'test' data of a case backup zipped over PM/direct message I can test internally for you to verify that this does restore.