Imagine painting a mirror and then beaming a sun-like spectrum at it. Will that still reduce the heat-producing IR wavelengths passing right through to the other side? What I'm thinking is, if you papered the ceiling with mylar material and painted it, could that improve insulation?

It depends on the properties of the material. Some transmit thermal IR (polyethylene does) and it looks like mylar transmits a decent amount near 10-15 microns

https://www.researchgate.net/figure/IR-transmission-spectra-of-polymeric-substrates-Mylar-with-thickness-d-40-mm-1-and_fig2_327932890