On my light I have mounted a small mirror in front of the LED to reflect direct light back to the light's parabolic reflector, and added a Fresnel lens to shape the beam. It is satisfactory in practice. Perhaps you could design and print the mirror mounting in thin steel.

Krylon makes a reflective spray that might work if you printed the reflector. If you can print in transparent plastic you could even do the lens. This would be a pretty good project with broad application IMHO.