As the global focus on environmental conservation and energy stability intensifies, enhancing energy efficiency and mitigating pollution emissions have emerged as pivotal issues that cannot be overlooked. In order to make a multi-energy-coupled integrated energy system (IES) that can meet the demand of load diversity under low-carbon economic operation, an optimal capacity allocation model of an electricity–heat–hydrogen multi-energy-coupled IES is proposed, with the objectives of minimizing operating costs and pollutant emissions and minimizing peak-to-valley loads on the grid side. Different Energy management strategies with different storage priorities are proposed, and the proposed NSNGO algorithm is used to solve the above model. The results show that the total profit after optimization is 5.91% higher on average compared to the comparison type, and the pollutant emission scalar function is reduced by 980.64 (g), which is 7.48% lower. The peak–valley difference of the regional power system before optimization is 0.5952, and the peak–valley difference of the regional power system after optimization is 0.4142, which is reduced by 30.40%, and the proposed capacity allocation method can realize the economic operation of the multi-energy-coupled integrated energy system.