Might be your chargin system and or battery. When you flash teh brights the change in power might get to low and the headunit is forced to turn off. But when you have them hi beams on constant they power drop is corrected. Also, When you listen to the radio the headunit is not drawing as much power.

1999 Ford Contour Lx, 2.5L V6 ATX, 145,000 miles Pioneer DEH-P6400 H/U , Rockford Fosgate 351M 350 Watt Amp, Pioneer TS-W305DVC 12" Sub, Compustar 2WSS, Whelen CSP650 Strobe Pack with 6 strobes, Whelen Talon B/B LED Dash Light