This work proposes a T-RIS-equipped LEO satellite communication in cognitive radio-enabled integrated NTNs. In the proposed system, a GEO satellite operates as a primary network, and a T-RIS-equipped LEO satellite operates as a secondary IoT network. The objective is to maximize the sum rate of T-RIS-equipped LEO satellite communication using downlink NOMA while ensuring the service quality of GEO cellular users. Our framework simultaneously optimizes the total transmit power of LEO, NOMA power allocation for LEO IoT (LIoT) and T-RIS phase shift design subject to the service quality of LIoT and interference temperature to the primary GEO network. To solve the non-convex sum rate maximization problem, we first adopt successive convex approximations to reduce the complexity of the formulated optimization. Then, we divide the problem into two parts, i.e., power allocation of LEO and phase shift design of T-RIS. The power allocation problem is solved using KKT conditions, while the phase shift problem is handled by Taylor approximation and semidefinite programming. Numerical results are provided to validate the proposed optimization framework.