We study the influence of many-body interactions on the transport properties in a two-site charge Kondo circuit recently implemented in a hybrid metal-semiconductor double-quantum dot device [W. Pouse, Nat. Phys. 19, 492 (2023)1745-247310.1038/s41567-022-01905-4]. There emerge two principal types of interactions: (i) an intrinsic one, described by the Luttinger liquid model, and (ii) an induced one, which appears due to the coupling of the system to an Ohmic environment. Case (i) could be achieved if the charge Kondo circuit operates in the fractional quantum Hall regime, while case (ii) can be implemented via a finite number of open ballistic channels coupled to both the quantum dots. We demonstrate that the conductance scaling for the case of strong and weak interdot coupling is fully determined by the effective interaction parameter, which is the combination of the fractional filling factor ν=1/m and the number of transmitting channels. Furthermore, we predict that the fractional filling factor ν defines a universal Kondo scaling in the vicinity of a special triple-quantum critical point featured by the emergence of a Z3 parafermion.