The bandgap at its heart -is- current mirrors -
jacked in a specific way, for effect.
You may not want an ideal-as-practical bandgap. The rolled-up drifts may want a nonzero TC to
compensate. PTAT, sure, but -what- proportionality?
That says you want to find "what makes it so" for the oscillator care-abouts.
If you have a reference voltage provided then a feedback unity gain buffer can get you "fixed"
currents using a FET follower with source resistance, Isink(t)=VREF(t)/RSET(t). Then mirrors and switches to get you up, down charging current at the timing cap.
Schmitt hysteresis band tends to be real sloppy as it's a 4-transistor tug-of-war with discrepant sizes, sometimes "history effects". A very not-great "reference" to hang the rest of it on. If you're going to the trouble of a true bandgap then you could do better for base reference stability.