How I would proceed if you require a reasonably accurate answer is as follows.
1) The free discharge is your control section. The normal equations for the level of the water do not apply in this region as they assume the streamlines are straight and parallel with the base of the channel. As the water goes over the dicharge, they are clearly not. I tend to use energy level rather than actual level in these circumstances. Work out the energy level and depth for critical flow for the channel width and flow rate you have.
2) Calculate the energy gradient and calculate the energy level a small step upstream of the free discharge. From this calculate the water level at subcritical flow. This is easiest to achieve using the Newton method of iteration, starting withthe energy level. The subcritical low level will normally converge within 5 iterations.
3) Keep repeating step 2 using the new level and moving upstream until the changes in depth at each stage are small. Then increase the step size.
With a flat channel, there is no normal depth. The depth will continue to increase as you progress upstream.
This method will not provide you with the perfect profile of the water. Near the discharge the water surface is lower than calculated. As you move upstream the actual profile starts to correlate with the calculated profile.