Variable LDO implementation - capacitor size
For the bypass cap on vout they say this:

VOUT: Regulated output voltage. A bypass capacitor is not required at
VOUT, but may be desired for good transient response. The bypass
capacitor must not exceed a maximum value in order to insure the
regulator can start.

They then say this:

A capacitive load on the regulator’s output appears as a short-circuit
during start-up. If the capacitance is too large, the output voltage
does not come into regulation during the initial tON period and the
UCC383 enters pulsed mode operation. The peak current limit, tON
period, and load characteristics determine the maximum value of output
capacitor that can be charged. For a constant current load the maximum
output capacitance is given as follows:

$$ C_{out(max)} = (I_{CL}-I_{LOAD}\frac{t_{on}}{V_{out}} $$
So choose how long you want the part to turn on with the first equation, then calculate the maximum capacitance and choose a value of capacitance lower than Coutmax.