Compared to traditional robotic systems, small-scale robots, ranging from several millimetres to micrometres in size, are capable of reaching narrower and vulnerable regions with minimal damage. However, conventional small-scale robots' limited maneuverability and controlability hinder their ability to effectively navigate in the intricate environments, such as the gastrointestinal tract. Self-propelled capsule robots driven by vibrations and impacts emerge as a promising solution, holding the potentials to enhance diagnostic accuracy, enable targeted drug delivery, and alleviate patient discomfort during gastrointestinal endoscopic procedures. This paper builds upon our previous work on self-propelled capsule robots, exploring the potential of nonlinear connecting springs to enhance its propulsion capabilities. Leveraging a mathematical model for self-propelling robots with a von Mises truss spring, which is verified using a finite element model, we investigate the effects of negative stiffness and snap-back within the nonlinear structural spring on the robots' propelling speed. Our analysis reveals that the negative stiffness of the von Mises truss can significantly reduce the sensitivity of the propelling speed to excitation frequency. As a result, the capsule robot exhibits a remarkably wider operational band where it maintains a high average propelling speed, surpassing its linear counterpart. This work sheds light on the potential for developing customised nonlinear structural systems for diverse scenarios in small-scale robot applications, opening up new possibilities for enhanced functionality and maneuverability in various biomedical applications.