PCL-based biodegradable shape-memory polymers (SMPs) are limited in strength, which restricts their practical applications. In this study, a series of novel SMPs, composed of poly(ethylene terephthalate) (PET), poly(ethylene naphthalate) (PEN), and poly(ε-caprolactone) (PCL), were synthesized and cross-linked using planar (benzene-1,3,5-tricarboxylic acid, BTC) or non-planar (glycerol, GC) cross-linkers via the one-pot method. The influence of different kinds of cross-linkers and hard segments of copolyesters on the thermal properties, crystallization behavior, mechanical properties, shape-memory performance, and degradability was investigated by FT-IR,1H-NMR, DSC, DMA, TGA, XRD, tensile test, intrinsic viscosity measurement, and in vitro enzymatic degradation test. The results indicate that the tensile strength of the copolyester can be significantly improved from 27.8 to 53.2 MPa by partially replacing PET with PEN while maintaining its shape-memory characteristics. Moreover, a small amount of cross-linking modification leads to higher temperature sensitivity, improved shape recovery rate at third round (Rr(3) = 99.1%), and biodegradability in the cross-linked PET/PEN/PCL shape-memory polymers. By changing the crystallization morphology and cross-linking forms of the material, we have developed a shape-memory polymer with both high strength and a high shape recovery rate, which provides a new strategy for the development of shape-memory materials.