Saturation Point of a magnet given some Parameters
Yes, there is a model that can help you relate all of the parameters of a magnetic hysteresis curve. There are several papers describing it and numerous codes available to help you fit the data to the model (some written in python).

The model is called the Jiles Atherton model

There is also a good description here on fitting the model.

If you need time series fitting or something else there are other papers if you search Jiles Atherton model or inverse Jiles Atherton Model.

I'm not aware if the parameters will give you \$ H_c\$ directly but you could find that once you fit the model easily.

This is another paper Jiles–Atherton Magnetic Hysteresis Parameters Identification that will help you fit the parameters to the model, its not trivial, but neither is nonlinear parameter identification.