Are blockchains inefficient if you’re not using proof of work? I thought it was the latter that was a source of inefficiency, and you don’t need PoW on a permissioned network.

If you're not using proof of work, blockchains start to look more like distributed storage with an audit trail and immutable row level db constraints

- if it's a Proof of Authority is like a way to deal with who has access to write to the db

- the distributed consensus part is well researched (Raft, Paxos etc)

The only advantage that I can think of is interoperability: you will be able to communicate with the public blockchain for point in time "backups". Also, building your internal smart contracts you can use an existing language like Solidity.

Even if you use some other consensus mechanism they're still inefficient. All full nodes in the network still have to propagate, validate and store all transactions forever.

This is indeed much more inefficient compared to a centralized solution and it's very difficult to create a decentralized network that can handle the amount of transactions that VISA processes for example.

For sure, doesn’t seem like there’s a need to solve for adversarial Byzantine style attacks here. Even a permissioned proof of authority blockchain is overkill for this (which is most likely what Accenture is building/configuring for them)

Maybe the blockchain is meant for "offline" transactions, when neither part is in contact with the Main Server.

Disclaimer: I dont know

Blockchains are highly inefficient for decentralized consensus as well.

But they are the most efficient known solution to the problem. Efficiency is relative.

No, they are the most efficient known solution to global consensus for double-spend. There are alternatives that relax the constraints using extra information such as connections in the network to get a relatively good approximation of the global consensus. See Safecoin, for example.