If your ETC contract wallet only requires one signature, you should be able to interact with it on MyEtherWallet.com or with Parity where you can load the JSON of your owner wallet. If your contract wallet requires two or more signatures, I'm not sure how to confirm the transaction but would love to know if anyone else does!
I finally figured it out! Turns out to be fairly simple and fast. Just need to run Parity in ETC mode (Target = "C:\Program Files\Parity\Parity\parity.exe" --chain=classic --geth), wait for it to sync, and then use the standard Ethereum Wallet (Mist) on top of that. The Ethereum wallet connects to the Geth that Parity is using and since it is in ETC mode, Ethereum Wallet is transacting on your ETC. Easy Peasy. This YouTube video shows how to do it well: youtube.com/watch?v=peAnJ3rRn04