Some advise seen is to transfer BTC to new addresses, then filter the coins, bit concerned about bleeding transaction costs there, or is this insignificant?
The transfer costs should be small if your BTC is not from a lot of inputs, however, I would not do this with Multibit Classic, since it's buggy and the risk to lose your money is big.
You should first export your private key and import it into Electrum, this way you'll be safer and you can also have control on your fees.
Then indeed, a good practice is to move your BTC to a new wallet, sweep the old wallet's Bitcoin cash and move those out too. I used coinomi for this, with this tutorial https://bitcointalksearch.org/topic/how-to-get-bitcoin-cash-from-many-walletsoftwareservices-with-coinomi-2062488