The phase transitions of one dimensional correlated Fermi gases in a transversely driven optical cavity, under the umklapp condition that the cavity wavenumber equals two times of Fermi wavenumber, are studied with the bosonization and renormalization group (RG) techniques. The bosonization of Fermi fields gives rise to an all-to-all sine-Gordon (SG) model due to the cavity-assisted non-local interactions, where the Bose fields at any two spatial points are coupled. The superradiant phase transition is then linked to the Kosterlitz-Thouless (KT) phase transition of the all-to-all SG model. The nesting effect, in which the superradiant phase transition can be triggered by infinitely small atom-cavity coupling strength, is preserved for any non-attractive local interactions. The phase transition occurs at finite critical coupling strength for attractive local interactions. Nevertheless, the critical dimension of the KT phase transition is also 2 like that in an ordinary local SG model. Our work provides an analytical framework for understanding the phase transitions in correlated intra-cavity Fermi gases.