Multiblock copolymer macrochain-transfer agents (MBP-CTAs) can be used to synthesize MBPs in one pot via reversible addition-fragmentation chain-transfer (RAFT) polymerization. MBP-CTAs can be efficiently synthesized via a copper-catalyzed azide-alkyne cycloaddition reaction. However, traditional purification methods cannot sufficiently purify MBP-CTAs, resulting in the failure of RAFT. In this study, it was discovered that using a supported catalytic system, namely, Cu@A-HNTs, inhibited the connection between Cu and S. The residual Cu content in the MBP-CTA was successfully reduced to a low level (7.88 µg/mg). Then, hydrophilic and hydrophobic MBPs were synthesized through RAFT polymerization using the MBP-CTAs. Subsequently, multiblock copoly(methyl methacrylate-b-butyl methacrylate-CTA)s (PMMA-b-PBMA-c-CTA)s were successfully prepared. The MBP-CTAs and MBPs were characterized using ultraviolet–visible spectroscopy, 1H nuclear magnetic resonance, size-exclusion chromatography, matrix-assisted laser desorption/ionization-time-of-flight mass spectrometry, differential scanning calorimetry measurements and uniaxial tensile tests. The results of performance testing of PMMA-b-PBMA-c-CTAs demonstrated that their fracture stress and strain significantly increased comparing with that of PMMA-b-CTAs.