Once the field ("the field", not "the field under control of the Safety Car", just "the field") has completed lap under FCY, there is no need to "re-grid", as the order of the "last scored lap" is exactly how they will arrive in Pit Lane under a BFA. If a Safety Car is dispatched, there are relatively few circumstances where the field hasn't completed a full lap under FCY by the time the Safety Car would make it back to Pit Lane (illuminated or not) and crosses S/F. No re-grid necessary (ibid).
The steward has 4 options:
1. Original grid
2. Running order on last lap scored
3. Restart from a scoring tape (which implies the last lap scored before the FCY was declared) and would be running order
4. Restart from a lap chart (which implies the last lap scored before the FCY was declared) and would not be running order.
Only one of those options does not require regridding the cars.