1) Using a VPN connect to shapeshift.io and use bitcoin to buy Monero and store in your local Monero wallet
2) Again using a VPN connect to xmr.to and sell your Monero for bitcoin and send to a local bitcoin wallet.
Why Monero? It intrinsically has a higher degree of privacy than Bitcoin due to the Cryptonote protocol used. It will be impossible to link the bitcoin you received back from xmr.to with the bitcoin you sent to shapeshift.io
Good idea to use Monero, as it does have built-in stealth features. But be careful about relying too much on VPNs for anonymity. VPN's provide privacy but not anonymity. Whoever runs your VPN has direct access to your true IP address, and you can only take their word that they won't share that info with third parties.
If I were using your method I would use tor instead of vpn, if shapeshift and xmr are tor-capable
Both shapeshift.io and xmr.to work fine over Tor. You just have to whitelist the sites in Tor browser so they can execute scripts.
Change into LTC, wait maybe a week or so so that their coins are already probably replaced, then change back to BTC.
As time consuming and laborious as this sounds, it may work.
If you don't like waiting, Bitmixer.io is legit.
That is the reason to use Monero. No waiting needed. I do agree it is good practice to change bitcoin to Monero using shapeshift then back to bitcoin using xmr.to. The transaction trail is completely broken. You can always use the taint analysis tool at blockchain.info to check.