There is a switch that is located at the base of the e- brake handle assembly. Either it's out of adjustment, or there is crap preventing handle to to lay all the way down. Remove 4 screws, two near seat belt retracters and two at the front near base of seat (move seat back and forth for easier access to screws). Pull up e- brake to help get center console out.
I'd suggest checking the brake fluid level. The light that gets turned on by low brake fluid level is the same light as the e-brake light. It sounds to me like the fluid level is borderline, and acceleration causes the fluid to move just enough for the float inside the reservoir to set off the light.
Iknow from personal experience that the floating sensor inside the M.C. reservoir can sort of hang-up in its 'well', and not drop down as it should. So at least give it downward push with your finger to see if its moving free.