Alkynyl-terminated polyethylene oxide−tetrahydrofuran (ATPET) and glycidyl azide polymer (GAP) could be linked through click-chemistry between the alkynyl and azide, and the product may serve a binder for solid propellants. The effects of the energetic plasticizers A3 [1:1 mixture of bis-(2,2-dinitropropy) acetal (BDNPA) and bis-(2,2-dinitropropyl) formal(BDNPN)] and Bu-NENA [N-butyl-N-(2nitroxyethyl) nitramine] on the curing reaction between ATPET and GAP have been studied. A diffusion-ordered nuclear magnetic resonance spectroscopy (DOSY-NMR) approach has been used to monitor the change in the diffusion coefficient of cross-linked polytriazole polyethylene oxide−tetrahydrofuran (PTPET). The change in the diffusion coefficient of PTPET with A3 plasticizer is significantly higher than that of PTPET with Bu-NENA. Viscosity analysis further highlighted the difference between A3 and Bu-NENA in the curing process—the curing curve of PTPET (A3) with time can be divided into two stages, with an inflection point being observed on the fourth day. For PTPET (Bu-NENA), in contrast, only one stage is seen. The above methods, together with gel permeation chromatography (GPC) analysis, revealed distinct effects of A3 and Bu-NENA on the curing process of PTPET. X-ray Photoelectron Spectroscopy (XPS) analysis showed that Bu-NENA has little effect on the valence oxidation of copper in the catalyst. Thermogravimetric (TG) analysis indicated that Bu-NENA helps to improve the thermal stability of the catalyst. After analysis of several possible factors by means of XPS, modeling with Material Studio and TG, the formation of molecular cages between Bu-NENA and copper is considered to be the reason for the above differences. In this article, GAP (Mn = 4000 g/mol) was used to replace GAP (Mn = 427 g/mol) to successfully synthesize the PTPET elastomer with Bu-NENA plasticizer. Mechanical data measured for the PTPET (Bu-NENA) sample included ε = 34.26 ± 2.98%, and σ = 0.198 ± 0.015 MPa.