Microwave inverse scattering imaging (MISI) is a crucial computational technique in microwave nondestructive evaluation and near-field microwave sensing systems. However, quantitative reconstruction of high-contrast targets remains a formidable challenge due to severe multiple scattering effects and the inherent ill-posedness of electromagnetic inverse problems. To overcome this fundamental bottleneck in computational microwave imaging, this paper proposes an alternating optimization framework based on cross-correlated physics-informed neural network (Alt-CC-PINN). This architecture deeply decouples the evolution of the microwave physical field from the neural-network-based dielectric parameter inference, replacing traditional joint optimization with a hybrid alternating engine. Specifically, the method employs an analytical Polak-Ribière conjugate gradient (PR-CG) algorithm driven by a cross-correlated loss to optimally update the contrast sources, and deploys batched zero-padded 2D-FFT to ensure high computational efficiency. Subsequently, a deep learning optimizer is utilized to update the continuous neural representation. Extensive validations based on simulated and measured data demonstrate that Alt-CC-PINN effectively overcomes the local minima problem in high-contrast and low-signal-to-noise-ratio (SNR) environments. It exhibits superior reconstruction fidelity and robustness under the frequency-hopping probing strategy, providing a powerful and reliable computational electromagnetic solver for practical microwave imaging systems.