How to apply Kirchoff laws to form differential equation for RC circuit with a variable applied 2nd voltage?
For example, if we consider the left-hand side 'loop', wouldn't Kirchoff's voltage law give us:
V2 = voltage across capacitor

thus eliminating all time dependence?

Correct, assuming the V2 source has no time variation.
If you connect an ideal voltage source across a capacitor, then nothing else in the circuit affects how it charges or discharges.