I'm certainly no expert on this either but let me explain my thought process on this...

If you make a withdrawal from Bittrex, then open an old version of the wallet and let it sync the withdrawal from Bittrex should show up since they're on the same fork.  At the same time, the old wallet is going to mint millions of coins that are invalid on the new fork, and all of this will be stored in the wallet.dat file.

Once everything is synced and the withdrawal from Bittrex (valid coins) and the minted POS coins (invalid coins) are in the wallet.dat file, and you open up the newest wallet, the valid coins should show up in your new wallet and the invalid coins should be rejected.  That's the whole point of the hard fork, right?