Airtable field formulas can’t touch files, and they also can’t affect base or table names. You could write a formula to create the name, but you’d have to copy it out of Airtable and apply it to the exported CSV file manually.
You might be able to use an integration service (Zapier, Integromat, etc.) to change the name prior to export, but I don’t know enough about either of those systems to know if they are granted access to modify base names. Same goes for Airtable’s own API.
If you opt to script something you run on your machine after export, I recommend using Python, as it’s available on all platforms, easy to install, and a relatively easy language to pick up.