In the original contest announcement comments, someone mentioned that TI had a great opamp guide, so I decided to check it out.
Inside, it too had a section on making a triangle wave generator, which is a fundamental piece to making a PWM.
The output of this circuit gave a much better square wave feeding into the integrator, and actually went rail to rail. However it still required a dual power supply.
The first opamp is a Schmitt Trigger circuit used to make a square wave. The square wave is then integrated by the second opamp circuit, which produces a triangle wave. The integrator is an inverting type, and its output is then fed back to the Schmitt Trigger to cause it to toggle as its thresholds are crossed.
All that was left to do was to have the triangle output feed into a variable threshold detector and we have PWM action! Followed by making all this work on a single supply.