No the exemption only applies to a new business and the LLC was already in existance.

Similar issue:  I have a single-member LLC that elected to be taxed as an S-corp mid-year 2021 so there's a Sch C for Jan-June and an S-corp for July-Dec, same EIN for the full year and the legal entity is still an LLC.  The single-member LLC paid the minimum tax in April 2021 of $800 and an LLC fee of $2,500 in June.  The FTB is saying the S-corp. owes an additional $800 min tax for the short period July-December?  Doesn't seem correct. I recognize the first year S-corp. minimum tax exemption is for new businesses only (your situation above) but this is the same ongoing business and the FTB is saying the 2021 $800 minimum tax is due for the SMLLC and again for the S-corp. for the same year.  Any guidance?

The FTB is correct. This is why you shouldn't change an entity in the middle of the year.