In the original Lamor equation, gamma^6 is used. Your approach optimzes this by using momentum instead of speed. 
However:
beta(t_i) = momentum(t_i) / ( mass * SPEED_OF_LIGHT * gamma(t_i) )
Is the time dependence of gamma in the vector part considered in your derivation?
I am not sure, if we can consider gamma to be constant between two consecutive time steps.