Including the {C special character is not working because that is the FNC1 code for the Epson printer. The special character for encoding CODE C is actually {1.

Also, as Terry Warwick alluded to, the font setting does not affect the printed barcode. I believe what you mean by Font A, Font B, and Font C are actually Subset A, Subset B, and Subset C which you indicate you would like to use by adding the appropriate special character above. When switching subsets within a barcode string, you should also include the Shift character, {S.

Try {APQR123X{S{11122331807110011223344

Can anyone point to documentation that unravels how Epson do font-switching in Code 128?

https://files.support.epson.com/pdf/pos/bulk/tm-i_epos-print_um_en_revk.pdf