If you care to use bitcoin-qt (or bitcoind) wallet, then a simple solution is to be offline while "sending" the money.
The outgoing tx will still show up in your transaction history (as an unconfirmed one) - you can check its TxID there.
Then (still being offline) use "getrawtransaction" RPC command to get the raw transaction data.
Now use Tor to load this webpage:
https://blockchain.info/pushtxAfter it loads, paste the raw transaction data into the box, press the button - and voila.
And now you can get your wallet back online.
Another method is to setup the node to connect through your Tor proxy.
That should also keep safe the IP of where your spending txs originate.
The only problem is (as someone pointed out) that Tor connection is much slower, but you only need to use it while actually spending the money.
Syncing up block-chain without Tor only exposes a fact that you are running a bitcoin node, but tells them nothing about its financial activity.