To export data from Phabricator, I would need to authenticate with Conduit as the omnipotent user. Is that somehow possible? Can I create an API token that belongs to the (or an?) omnipotent user?
No, that shouldn’t be possible - the omnipotent user is only supposed to be available directly from code.
You should be able to write a script to run server-side which will export in similar format as the conduit methods, except that this can run as omnipotent.
So instead of making HTTP requests I would run
./bin/conduit call --method ... --input - and that would be omnipotent?
maybe? give it a try, it might.