Particle and potential wells are in the framework of quantum mechanics. In this framework one cannot be talking of potential wells arbitrarily changing the particle's energy, because the energy is strictly defined by the solution of the quantum mechanical equation for the given potential.
What happens to a particle as it gains an infinite amount of energy? Does it stay inside of the infinite well?
Here is an example with specific boundary conditions of an infinite potential well using the time independent Schrodinger equation for the solutions.
The particle can be in one of these states
where n can go to infinity . The energy is on the y axis . Taking a limit of n to infinity , a level exists at at each step, since the solution is a periodic function.
Does a particle with infinite energy escape an infinite well?
For this model, no. It will be caught in one specific value of n. There is no "outside" in this model.
The issue is to accept that particles and potential wells belong to the quantum mechanical regime and the models have to follow specific rules.
Do I need to define the rates at which the potential of the walls go to infinity, or the rate at which the particle's energy goes to infinity?
One may model infinite potential wells in different ways, also time dependent, BUT the possible energy states of the particle are defined by the potential well and the boundary conditions, one cannot change independently the energy of the particle.