There are several polyester-based insulation products available; Bradford makes polyester based "Polymax" batts which were formulated to reduce the dust and skin irritation produced by traditional fiberglass batts. The garment industry has a regulating body that, among other things, classes apparel fabrics according to ignition hazard. Polyester has a high rating which makes it safe in relative terms. ASTM E84 tests interior building finishes for flame spread after ignition, and polyester fabrics are often rated Class A, which means relatively low flame spread. Most large drapes you might see in hotels are polyester or have a generous amount of polyester fiber to allow them to hang with good sheer and little wrinkling.
Almost all building insulation is required to be covered by an ignition barrier, like gypsum board, in conventional light frame or steel frame and infill. It is far more likely that ignited room contents, furnishings and interior finishes will reduce occupant escape probability than insulation located within a stud bay due to this fact.
The two issues I see with using scrap fabric for stud cavity insulation are lack of homogeneity and consolidation. The fabric scraps will have to be spun or ground to create a homogeneous, air-entrapped material to provide an efficient insulation, and then supported to avoid consolidation at the bottom of the stud bay and air gaps at the back of the interior wall finish. I have witnessed the removal of interior finishes on hundreds of walls during my career, and every single loose-fill application and most of the batt applications were in support failure at the time of removal. Many failures for loose-fill occur after only 2-3 years; batts fail usually after 5-8 years if improperly supported.
The three best practices for insulation design are continuity, support and location within the wall section. So, for example, low R-value well-supported continuous insulation installed dead tight against the back of interior wall board will provide more comfort and energy savings than discontinuous high R-value poorly supported insulation installed with an air gap between the interior wall board. If you decide to use the scrap fabric, I'd be interested to hear how you end up installing it.