Are Bluetooth batteries required on cordless tools to be able to use the cordless dust extractor?

No, Bluetooth batteries are not required. Cordless tools with no Bluetooth batteries can be used and the dust extractor can be turned on and off at the tool by using the Cordless Bluetooth Remote (202098) which is included with the Dust Extractor

https://www.festool.com.au/bluetoothremote-control-for-additional-hoses-202098.html