Flask: View, model and business logic segration
If you need to unit test the code separate from the View then you should defiantly separate it into another module or class.   
As there seems to be three parts to your business logic then I would say starting by splitting the view into three functions of a module seems a good place to start.
Please help me how to solve following task in "pythonic" way:
There are several model classes, which are mapped to the DB with the help of SQLAlchemy.
There is a Flask view, which handles the "POST" request. 
The business logic of this method contains complex logic which includes:

Getting input parameters from input JSON
Validation
Creation of several different models and the saving to database.

Is it a good idea to leave this logic in the "View"? Or it would be much better to separate this this logic into different modules or classes, for instance by introducing business logic class?