What kind of equipment would I need to apply nonpermanent extension wires to test points of a PCB so I can access them via a microcontroller?
This is one of those things where what you think you want and what is possible to do are about 10,000 meters apart. 

It is possible to make a jig that has spring loaded pogo pins. Pins with sharp points can be made to line up with small test points on one side of the PC board. Generally the board is guided in place on the jig via tooling posts that line up in holes in the circuit board. For longer term use a clamping mechanism is needed to hold the board down in place on the spring loaded pins. 

The pogo pin fixture is going to take up a good amount of height so give up the idea of it being portable and fittable inside the existing housing. It is also nearly impossible to build a successful pogo pin fixture unless you have the original design data for the circuit board. Gerber files would be a minimum. You also need a precision drill fixture that can drill perfectly straight holes down into the material that holds the pogo pins in place. That material needs to be a good thickness to hold the pins in a sturdy manner. Expect a pogo pin fixture to take up about 2 -> 3 cm of height depending upon which pogo pins you choose.

For trying to fit the connections inside the existing case you are going to have no choice but to solder the wires. Give up trying to think otherwise. Wire the 15-20 wires over to a small connector that you might choose to poke through a hole that you Dremel out in the existing housing. 

Keep in mind that once you start down the road of hacking into a pre-existing product that it becomes dedicated to that effort. Erase thoughts of trying to keep your hack platform pristine. If you need the product for normal use then go purchase another one for that purpose.

Just for reference here are a couple of pictures of a small pogo pin fixture that was made to allow programming of an AVR microcontroller via test points to its SPI programming port directly on its product circuit board. In this case the programming operation took only a few seconds so the product circuit board was simply held in place on the tooling posts. Note in the side view that the polycarbonate that holds the pogo pins is ~9.3mm thick.