Global Navigation Satellite Systems (GNSS) aided Inertial Navigation System (INS) is a fundamental approach for attaining continuously available absolute vehicle position and full state estimates at high bandwidth. For transportation applications, stated accuracy specifications must be achieved, unless the navigation system can detect when it is violated. In urban environments, GNSS measurements are susceptible to outliers, which motivates the important problem of accommodating outliers while either achieving a performance specification or communicating that it is not feasible. Risk-Averse Performance-Specified (RAPS) is designed to optimally select measurements to address this problem. Existing RAPS approaches lack a method applicable to carrier phase measurements, which have the benefit of measurement errors at the centimeter level along with the challenge of being biased by integer ambiguities. This paper proposes a RAPS framework that combines Real-time Kinematic (RTK) in a tightly coupled INS for urban navigation applications. Experimental results demonstrate the effectiveness of this RAPS-INS-RTK framework, achieving 85.84% and 92.07% of horizontal and vertical errors less than 1.5 meters and 3 meters, respectively, using a smartphone-grade Inertial Measurement Unit (IMU) from a deep-urban dataset. This performance not only surpasses the Society of Automotive Engineers (SAE) requirements, but also shows a 10% improvement compared to traditional methods.