In this study, waste polyethylene terephthalate (PET) drinking water bottles were upcycled into 3D-printing filaments to reduce the carbon footprint of a material extrusion process. The potential improvement in recycled PET (R-PET) properties was investigated using two reactive additives: a multifunctional chain extender (CE) and a peroxide free-radical initiator. It was found that both neat and modified R-PET filaments with the lowest additive content could be printed to 100% completion at a lower temperature of 250 °C compared to virgin PET (V-PET) filament, which required 260 °C. CE modification enhanced the melt strength, evidenced by a higher low-frequency storage modulus (G'), and provided stable filament dimensions. However, the 3D-printed specimens from CE-modified R-PET exhibited poorer tensile toughness (0.67 MPa) and greater warpage (3°) compared to those made from neat R-PET (1.0 MPa and 2.30°). In contrast, filaments produced from peroxide-modified R-PET resulted in specimens with the highest tensile toughness and the lowest warpage, with an optimal peroxide content of 0.25 wt%. Higher printing temperature (270 °C) and bed temperature (80 °C) maximized tensile toughness to 2.2 MPa and resulted in a milder warpage angle of 1.5°.