I don't think the app supports it as I wanted something similar but to use my Bluetooth headphones to not bother people around me.
Issue is with lag for audio between vlc, the chromecast and the Bluetooth device
After months of trying 3rd party apps and such I settled on getting a chromecast 4k with google tv which isn't too expensive and sometimes goes on sale
You can get the HD version but I ended up returning it and getting the 4k one for the faster processing
https://store.google.com/us/product/chr ... n-US&pli=1
Reason this worked for me is apps can be installed directly on the chromecast including vlc. Also you can pair a Bluetooth device with it for audio output.
Also in your case you can setup a media server like plex on a computer and stream from it directly from vlc on the chromecast
In terms of apps other than vlc that support chromecasting and external audio on Bluetooth I used to use this one but not all files are supported and also you have to manually adjust audio sync so it matches video on the chromecast
https://play.google.com/store/apps/deta ... t.webvideo