We have been discussing this very flag on my online flag historians group of late.
The flag was probably taken from one of the trapped blockade runners in Mobile after the Blakely battle itself. There has been some attempt to tie this to Culpepper's SC Battery but the flag's sheer sixe, 7 1/2 by over 10 feet, is far too large for a unit color. This is much more in line with sailing flags.
When Mobile finally fell, there was loads of souvenir hunting by the Federal troops and that is where this flag was probably taken.
The CSS Dixie used a red flag with white crescent and white star - the same flag being hoisted over the US Customs House in Charleston when it was taken over. It was later presented to the captain of the Dixie.
As for Palmetto flags themselves, I have a thick file of this pattern showing up all over the place in the secession era. In fact, this flag, and not the lone star/Bonnie Blue flag, is the REAL flag of Southern secession and far more examples of them exist than any lone star flag pattern that I have been able to find in the period newspapers.
Palmetto flags were hoisted in Chicago, Philadelphia (who also had a pro-South newspaper for a time called the Palmetto Flag), Connecticut, Yale University, St. Louis, all over the South and even as far west as Nebraska Territory!
Most were blue, while others were white. I would have to pull it out to see if there were any red Palmetto flag used.
This flag only failed to get its proper recognition as THE Southern flag of secession because it did not have a hit record to go along with it.