Sorry for the delay. I thought that we were finished!
My previous post shows the formula with a common B and beta and conditional parameters on Category levels. What else do you need? The script in my first reply automates building such a formula. I can't understand what else you might require from a model stand-point.
Are you asking about how the Nonlinear platform solves such a problem (estimate the parameters)? It uses a numerical optimization procedure and a loss function to monitor. It uses the given starting values and then varies them to improve the result of the loss function (minimization). The default loss function is sum of the squared errors, or least squares. It monitors several measures of change to determine when convergence is achieved and it stops.