Calculating optimal path(s) to increase probability of a classification
This is a multivariate optimisation problem. You have a function f(X) that returns an output that you want to maximise - Maximising the probability of belonging to class. That function f(X) is just your model. 

You also mention "constraints", and again this is standard constrained optimisation territory. 

The problem itself may be hard to solve, but the framework seems to be that. 

Have a look at scipy optimise here