Use a VPN to send transactions from your place. Use the bitcoind RPC as the public interface.
If the transaction is signed then there is no reason to encrypt... it will be public anyway.
If you insist on Tor, rent a cheap VPS. Install the Tor hidden service software and configure the ssh (port 22) interface. Use iptables to close all inbound ports. At your end, get a copy of Tails, connect to the .onion address using ssh, and run bitcoind. Use the bitcoind CLI to send commands.
Transaction will be public, sure, but, routing the broadcast over tor prevents anyone listening to your traffic realizing you sent it, just that someone, somewhere in the world sent it. I can already (And have already) done things like you recommend (Albeit, without the VPS, just two copies of bitcoind, one configured for tor, one not), it's just a huge pain to constantly jump between the two clients, one for creating/managing funds/transactions, one for actually broadcasting.
2. Go to https://blockchain.info/pushtx
3. ?
4. Profit! Er..Anonymity!
'Tis be a huge pain for every single transaction.