Computational imaging for hyperspectral images (HSIs) is a hot topic in remote sensing and imaging systems. The dual-camera compressive hyperspectral imaging (DCCHI) system has been successfully designed and applied in hyperspectral imaging. However, the corresponding reconstruction algorithms are not well developed. In this paper, under the DCCHI framework, a new reconstruction algorithm is proposed based on the collaborative Tucker3 Tensor decomposition. In actual HSI, similar nonlocal patches always have similar spatial-spectral structures, and thus, these nonlocal patches can share the same spatial and spectral factors in Tucker decomposition. To characterize the similarities simultaneously, the Tucker3 decomposition is used to model the 4-order tensor formed by the similar cubic patches. To keep the spatial structures in the reconstructed HSI consistent with the panchromatic image's spatial structures, we force the spatial factor matrices and the core tensor in the Tucker3 decomposition of the HSI to be identical to the spatial factor matrices and core tensor of the panchromatic image's Tucker3 decomposition. In addition, a spectral quadratic variation constraint is introduced into the spectral factor to characterize the band smoothness. To solve the optimization problem, an alternating direction method of multipliers (ADMM)-based algorithm is designed and each variable is separately solved. Experimental results on a public data set and the remote sensing image demonstrate the advantage of the proposed method.