Your ~20mA may only be average and not peak.
To support 100mA peak current, Assume hFE drops to 10% or Ic/Ib=10 for rated Vce(sat)
Since 1st NPN stage does not saturate , assume hFE=100
2nd stage PNP saturates assume hFE=10 worst case
Thus to support PNP output = 100mA peak, Ib2=10, Ib1=100uA
For 0.2V=Vce2 R load ~29 Ohm for 100mA peak
Rb series to PNP= 220 Ohm
Rb series to NPN= 22K
Not ratios 22k / 220R/ 29R ratios are 100, 8
As Vbat increases Vbe can be nelected then, and it approaches 100,10 for R ratios with respect to Load worst case min dynamic R
You might not have scoped the Vout when it was at peak current.
If not 100mA or 20mA but something in between, then calc worst case Rload and use the ratios of 100,8 x Rload for Rb1 Rb2