Thank you for these answers. So I'm encouraged that basically you're saying it's doable, at least in concept. I don't mind a few extra steps.
Some clarifications:
Actually, sweeping is recommended since when you export the private keys, you are at more risk of them being stolen. Sweeping makes sure that even if someone steals your private keys, there won't be any Bitcoin for them to steal. It won't work if you are offline.
I don't see there could be more risk of them being stolen.
I'm planning on exporting the keys from Qt and importing the keys to Armory as follows:
(1) Copy the Qt wallet.dat into the offline machine, fire up Qt, and use the command-line interface in Qt to export the private keys.
(2) Import the private keys to Armory - also on this same offline machine.
So... How could anything get stolen in this scenario?
Oh, I think I get what you're saying the risk could be: someone else could have a copy of those private keys from the Qt wallet. But it's been under my exclusive control this whole time, and besides, those private keys are only going to be used for a short time (in the new Armory "temp" wallet). Their funds will be immediately sent to the Armory "permanent" wallet, in step (2)(d) of my original post.
The problem is that bitcoin core uses compressed keys now and armory only supports uncompressed keys, so the keys from Bitcoin core won't be able to be swept or imported.
This is actually a really old wallet - I think it's about 3 years old. So I think it doesn't use compressed keys.
Possible alternative:
I guess I could also simply sign a raw transaction from Qt itself and then broadcast it to my new Armory offline "permanent" wallet, from a site like blockchain.info/rawtx. This would obviate the initial step of importing to Armory, into a "temp" wallet, as mentioned in my step (2)(a).
However, since I'm going to have to learn anyway how to offline-sign and broadcast transactions in the future using Armory, and since I haven't learned how to do that yet using Qt, I figured I might as well just get the private keys into Armory, and then have an all-Armory all-offline solution for getting the funds from Qt to Armory, without having to learn how to offline-sign raw transactions from Qt.
I would only do this alternative (offline-sign a raw txn from Qt, and broadcast from blockchain.info/rawtx) if the keys from Qt actually do turn out to be compressed. But this wallet is really old, so I think they're non-compressed.