Thanks for the pictures. They are actually very helpful.
It sounds like you cannot do that without the chime connector. If you connect the doorbell directly to the “1NOTE”, then the doorbell unit draws current through the chime solenoid and that makes it buzz. Is there a reason that you prefer not using the chime connector?
Now the key item is to determine, how the doorbell tells to the chime connector that it has to ring the chime. Probably there is a relay or something inside the chime connector and load modulator inside the doorbell to actuate the relay. It might worth testing with some relay how it works. Perhaps relay like
would work. You’ll need to connect the relay’s coil between the TRANS and doorbell and the normally open relay contacts between “1NOTE” and COMM.
If you have a current meter, you could perhaps measure the current draw of the doorbell (set the current meter to AC range, then connect one probe to “TRANS”, the other probe to doorbell (assumed here that doorbell’s other wire goes back to transformer). Then measure the current when the bell is not pressed and when it is pressed. The pressed time moment may actually be so short that the current meter does not detect that, but is at least worth trying.