The present study reports on the development by thermoforming of highly sustainable trays based on a bilayer structure composed of paper substrate and a film made of a blend of partially bio-based poly(butylene succinate) (PBS) and poly(butylene succinate-co-adipate) (PBSA). The incorporation of the renewable succinic acid derived biopolyester blend film slightly improved the thermal resistance and tensile strength of paper, whereas its flexural ductility and puncture resistance were notably enhanced. Furthermore, in terms of barrier properties, the incorporation of this biopolymer blend film reduced the water and aroma vapor permeances of paper by two orders of magnitude, while it endowed the paper structure with intermediate oxygen barrier properties. The resultant thermoformed bilayer trays were, thereafter, originally applied to preserve non-thermally treated Italian artisanal fresh pasta, “fusilli calabresi” type, which was stored under refrigeration conditions for 3 weeks. Shelf-life evaluation showed that the application of the PBS–PBSA film on the paper substrate delayed color changes and mold growth for 1 week, as well as reduced drying of fresh pasta, resulting in acceptable physicochemical quality parameters within 9 days of storage. Lastly, overall migration studies performed with two food simulants demonstrated that the newly developed paper/PBS–PBSA trays are safe since these successfully comply with current legislation on plastic materials and articles intended to come into contact with food.