Just to eliminate some obvious, make sure steering servo is plugged into channel 1, ESC in channel 2

The 1256 servo shouldnt be too much for the BEC its not super fast like the 1257 so it shouldnt have huge spikes.

The reason i say this is i have 1257 and 1258 servos, i run MMP's and Tekin RS's. The RS BEC has no trouble pulling any of these servos, but when i put a 1257 on the MMP i need to run a cap. No issues out of the 1258. Odd rite? But thats how it is and I need to run a cap when using a MMP/1257 setup.

Still if it is the servo pulling too much i would look into a receiver capacitor you can get for like 5 bucks over spending 30 on an external BEC.

I really like the MMP, great tunability, smooth, and handles lots of power. Its only downfall is it's weak BEC circuit.