Upgrading a Sport license to a Private license - is a new student license necessary to solo non-LSA aircraft?
You do not need to get another student pilot certificate.  The endorsement you need is not for student pilots.

70. To act as PIC of an aircraft in solo operations when the pilot does not hold an appropriate category/class rating: § 61.31(d)(2).
I certify that (First name, MI, Last name) has received the training as required by § 61.31(d)(2) to serve as a PIC in a (specific category and class of aircraft). I have determined that he/she is prepared to serve as PIC in that (make and model) aircraft. Limitations: (optional).
/s/ [date] J. J. Jones 987654321CFI Exp. 12-31-19

The answer lies in Advisory Circular 61-65.  Version F is the latest.