What can cause the output voltage of a buck converter to drop (apart from high current)

First, a voltage regulator whose voltage can be changed by a microcontroller is more typically called an digital to analog converter, or DAC. You can send the output of a DAC to a buffer to increase the current, but there will probably† need to be a DAC somewhere in your system.

Second, before trying to simulate something, you need to come up with a model. The simplest model for a battery is a voltage source. You've recognized that a perfectly stiff voltage source is inadequate, because battery output dips with increased current. A little research will indicate that the cause of this drop with increased current is internal series resistance; put a resistor into your model and you're good to go! More can always be done, of course - Take a look at this article for more detailed models:

The primary modifier in this example is a series resistor, so, rather than algorithmically sensing the current and decreasing the output voltage, you could simply add a resistor. A resistor will instantly respond to changes in voltage and current, without any complicated code on a microcontroller. If you find that you need to model transient behaviors, add a transient resistance and capacitance.

If you need to occasionally change the resistance, then use a potentiometer. If you need to do it while the system is running, use an in-line MOSFET to build a variable resistor or buy a digital potentiometer IC.

If you still need to measure the current, the customary way to do this is to add a small (insignificant) series resistance and amplify the voltage across this resistor. If you use only linear regulators with minimal quiescent current, you can put this sense resistor at the supply of your voltage regulator, alleviating the problems you mentioned in your question. The voltage will still be small, and it won't be referenced to 0V, so you'll need to amplify and condition it. The easiest way to do this is with a high-side current sense monitor IC.

† _{You can also do this with filtered PWM or by directly driving a switching regulator from your microcontroller, but those are both trickier than simply using a DAC to output a voltage.}

What is the frequency of the oscillation you observe?

If it is relatively low frequency, then the drop is probably caused just by energizing the TPS61222 output cap and inductor. You could fix that by adding more hysteresis on the comparator. Seems unlikely to me, however, because the super cap is so large.

If it is highish frequency, it could be caused by inductive and resistive drops in the path between the supercap and the input to the TPS61222 (including the internal resistance of the supercap itself, which is likely considerable). Again, adding more hysteresis on the comparator will help fix it. You can also improve the situation by using a bulk input capacitor right next to the inductor to hold the input to the TPS61222 in the face of the relatively large currents you will see at startup - this will work if you take Simon's recommendation and drive the EN pin of the boost converter instead of a load switch. That recommendation is likely a good one anyways, since it saves you a part.

Note that if you are running a load at the output of the boost, you may observe some sort of "oscillation" any time your load power exceeds the input power from the cell. The solar cell will charge up the super cap until the boost turns on, at which point the power drawn from the SC exceeds that going in and the voltage starts to drop. It continues dropping until the TPS61222 turns off / gets disconnected, and then the process starts over.