A little birdie told me that you can use https://counterwallet.io/ to create transactions and view your balance of Armory offline wallets, without ever revealing your private keys.
Whoa! I belive this is exactly what I was looking for. Thank you, njaard!
But do read the small print: It only works with addresses that have already been used to SEND bitcoins at least once, because they need the public key, and they get that from the blockchain. Of course if there is a market for it, they could easily make an interface reading it directly from the watch-only wallet, making their site equivalent to the online Armory computer (except for privacy issues, of course).