I can send BTC from that wallet, but I can only get the fee as high as BTC 0.0005, which I think is probably too low to get a transaction confirmed.
Does anybody know the best way to export this money to a different wallet so that I don't have to wait weeks for a transaction to clear due to lack of sufficient fees?
It clearly shows the steps required to export your private keys from MultiBit Classic. I'd recommend Electrum as a wallet, but you can use any wallet that allows importing of Private Keys.
Three things to note:
1. After you create the export file (DO NOT password protect this file, otherwise it will be encrypted), make sure you copy/paste JUST the keys and not the date/time stamps... Keys highlighted in red:
2. If you go for Electrum, make sure you are using the very latest (v3.0.5) version of Electrum, available here: https://electrum.org/#download (earlier versions have a critical security flaw)
3. The video shows importing to v2.9.3, however the "import" option in v3+ changed... so to import your private keys you MUST use the "Import Bitcoin addresses or private keys" option when creating the wallet:
DO NOT use: "Standard wallet" -> "Use public or private keys"... this will NOT work on v3+