Capacitor sizing for known current load

I would use an easier approach: the capacitor is only needed to act as buffer for the high-current spikes. Just get the duration of the (biggest) spike (t_duration) and the height of the (biggest) spike (I_max) from the data you got ("For the time period of interest, I have the load's current draw over time.").
The capacitor's voltage should never drop below 90%, so dv = 10% * Vs.

Use
$$i = C*dv/dt $$

$$ C = \frac{ I_{max} * t_{duration} } { 0.1 * V_s}$$

This will yield a bigger capacitor than needed, so the voltage drop will be quite less than 10% of Vs.

Another approach is to simulate the circuit is Spice, putting the data you have ("For the time period of interest, I have the load's current draw over time.") in a Piece-wise linear current source. Pick a starting value for C, like e.g. 100uF (or use the value given in the first approach) and tweak by running a few simulations.

If you really want to solve your approach, I would suggest iteratively picking values for C and evaluate the definite integral.