Using a high resistance pull down resistor
The exact set up I was envisioning was this. I use a very large pull
  down resistor (on the order of mega Ohms). The voltage drop across
  this resistor will be so large that it will essentially behave as a
  pull up resistor


This doesn't make a lot of sense. If you hook the resistor from the pin to ground it's a pulldown. The voltage at the pin is the same as gnd. By ohm's law, there is very little voltage drop across the resistor because such a small amount of current flows.

A typical configuration is shown below.

I don't really understand your idea, to be honest. It sounds like you're trying to have a big resistor that's both a pull up and a pull down, then connect a smaller  parallel with a switch? I'd need a schematic to comment further, but you really only need the standard pull up or pull down configuration. For a pull up, swap the position of the resistor and button.







simulate this circuit &ndash; Schematic created using CircuitLab