A novel scheme for a double closed-loop resonant fiber-optic gyroscope (R-FOG) employing a high-performance transmissive Kagome hollow-core photonic crystal fiber (HCPCF) resonator is proposed. We use specially designed Kagome HCPCFs and a meniscus lens module to form a resonator, whose finesse is 58.2 with a length of 5.6 m and a diameter of 13 cm; the theoretical sensitivity of the R-FOG is better than 0.05°/h. Based on the novel Kagome HCPCF resonator, a double closed-loop R-FOG is set up and the performance of the R-FOG system is experimentally studied. It demonstrates that white noise dominates in the output at the integration time of 200 s and a bias stability of 0.15°/h and angle random walk coefficient of $0.04^{\circ}\,{\rm h}^{1/2}$0.04∘h1/2 are achieved. Over a dynamic range of ${-}{100}^\circ {\rm /s}$-100∘/s to 100°/s, the scale factor nonlinearity of the FOG is 310 ppm, which shows significant improvement compared with the single closed-loop R-FOG system. The novel double closed-loop R-FOG proposed is quite feasible for tactical grade applications.