The older model used to have a setting for “ring sensitivity”.
Looks like your PBX equipment does not put a good ringing voltage out, or it does it in short burst instead of the international setting of one second on/two seconds off.
Put a regular phone on the PBX extension,ring it and see what ringing pattern and AC voltage you are sending out of the PBX.
Repeat the procedure with the PSTN line. You are bound to find that the voltage coming out of the PBX is diferent.
Also, the PSTN is almost always 20 cicles per second ringing, when on PBX could be as high as 50 cicles.
Just some troubleshooting ideas.