The integration of renewable and fossil energy in a Region Integrated Energy System (RIES) is recognized as an effective measure for carbon emission reduction. The emergence of carbon emission trading (CET) markets worldwide has raised the need to combine RIES optimal planning with CET to achieve carbon reduction targets. This study focuses on the synergetic competition of a cross-RIES system with low-carbon technology heterogeneity under the CET mechanism. A cross-RIES model is constructed, incorporating fossil Carbon Capture System (fossil-CCS), renewable Combined Cooling Heating and Power (renewable-CCHP), and cleaner production paradigms. An improved version of the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is used to propose a bi-level multi-objective optimal planning framework for the cross-RIES model, employing a grandfather permit allocation method. Comparative analysis is conducted between cross-RIES optimal planning with and without considering the CET mechanism, with a focus on environmental indexes such as carbon emission, carbon intensity, and marginal abatement cost. Sensitivity studies are also performed to explore the impact of key CET design parameters on the system's environmental performance. The simulation results demonstrate that the CET mechanism effectively reduces carbon emission, primary energy consumption, and carbon intensity by promoting synergetic competition among the low-carbon technology heterogeneity of RIESs. Furthermore, the environmental performance varies among the three types of RIES. In terms of CET parameter design, lowering the cap constraint, reducing the allowance allocation ratio of fossil-CCS RIES, and increasing the trading price contribute significantly to reducing carbon emission and carbon intensity, albeit at the expense of higher marginal abatement cost.