If it requires the id as part of the reference, I would look to set this using the created event fired by eloquent when a record is created, and update the reference then (so it will need to start as a null value).
https://laravel.com/docs/6.x/eloquent#events (or observers https://laravel.com/docs/6.x/eloquent#observers)
so in your event handler, you would just do something like:
$record->update([
'serial_number' => 'Bill19-20-' . $record->id
])
And the 19-20 you would just need to add based on the dropdown selected.