A coil former mfg once told me to spray some silicone in the threads to free them up a bit for the core to turn. be very gentle, they break so easy, try rocking back n forth never force.
Probably best to leave it alone
I'd like to help but this thread is too confusing to follow, plus too many pilots is no good either.
To make it easy for me and the next chap with the same issues, try to list the faults and the fixes, one by one in a summary
Learning is reading, like "does anyone know what pin the input comes into HA1196 to tell it to turn pin 13 to low"
I say read the datasheet perhaps, just like I would be doing since I do not know for sure, I'll guess 25mV of 19kHz pilot, How about you check to see what Hitachi data sheet says.
Good luck