Optimal procedures and appropriate designs of transmitting channels are required to preserve quantum correlations (QCs) in quantum teleportation schemes. In pursuit of this objective, a comprehensive investigation into the intricate dynamics of a quantum channel comprising two singular Cooper-pair charge qubits, mutually coupled and influenced by Josephson energy, has been undertaken. The use of the Gibbs density operator has provided valuable insights into the thermal quantum characteristics and teleportation fidelity of the system. Furthermore, this study underscores the critical interplay between the energy parameters and the utilization of quantum features to establish conducive conditions for the preservation of nonclassicality and the success of the teleportation scheme. Additionally, the study reveals that trace distance discord is the most resilient in the thermal output state, surpassing both concurrence and the degree of violation of Bell’s inequality. Notably, precise control of the input state amplitude and adept manipulation of the energy parameters have enhanced QCs and resulted in the teleportation fidelity exceeding the classical limit.