Biodegradable polyesters represent an advanced alternative to polyolefin plastics in various applications. Polybutylene adipate terephthalate (PBAT) can compete with polyolefins in terms of their mechanical characteristics and melt processing conditions. The properties of PBAT depend on the molecular weight, dispersity, and architecture of the copolymer. Long-chain branching (LCB) of the PBAT backbone is an efficient method for the improvement of the copolymer characteristics. In the present work, we studied branching agents (BAs) 1–7 of different structures in the two-stage polycondensation of 1,4-butanediol, dimethyl terephthalate, and adipic acid and investigated the composition and melt rheology of the copolymers. According to the results of the research, 1,1,1-tris(hydroxymethyl)ethane 2 and 3-hydroxy-2-(hydroxymethyl)-2-methylpropanoic acid 5 outperformed glycerol 1 as BAs in terms of shear thinning behavior and viscoelasticity.