To tell you the truth although I designed the schmitt trigger version I never tried it for real on the smps.
Things you can try
1) You can remove the input thermister and put a 5watt 47R resistor instead. To help a slower start up
2) Following my schematic for schmitt trigger there is a decoupling capacitor which is coming from the sample diodes (1N4148) and has a value of 220nF, You can rise the value slowly 1uF- 4.7uF until you get a start up without tripping on the surge current, Keep to the minimum possible otherwise during a short circuit your fets will not be saved.
I have put the setting to the minimum, I do not think that the soft start will solve it since when the source is protected by protection, for some reason and no matter how much capacitors are discharged, it starts again protected.
However, if I make a bridge between the base and emitter of the 5551, the protection is removed, leaving me to have output voltage. I proceed to disconnect the source, turn it back on and it turns on without any problem.
A colleague told me to implement the following:
Although I would not know well what values to place.
R, C y Tr
Greetings!