What is causing this step-up converter circuit to fail when introducing a transistor?
Your transistor doesn't have enough gain.

When the solar panel is dark, it looks like a resistor.  That's why the transistor turns ON when the solar panel doesn't have any light on it.

Because your SMPS regulator is a boost circuit, it requires more current with its input at 1.2V than your circuit would need if it was running directly from 3 cells (nominal 3.6V).

The combination of low gain and small base current means that the transistor can't turn ON fully.

You can try a couple of things:

1) put a resistor in parallel with the solar panel.  You will have to experiment with the value.

The downside of this is that now you are wasting solar energy in this resistor.

2) Use a P-channel low-threshold-voltage MOSFET instead of the transistor.  Do a search for a family of MOSFET called "Trench FET" - Digikey has several that are suitable.

You want the Rds-on resistance to be a low as possible and the fully-enhanced gate voltage needs to be lower than 1.2V.