The first problem is that this is an IP Office running in Basic mode. A Partner emulation, so it does have built in support for Avaya door boxes. It also does hotline dialing. No one in their right mind has that equipment sitting around to test and provide a solution.

On an IPO in Essential or Preferred mode, you would use a single line port and hotline dial to a User or Group. I reused some weird version of Avaya door box recently, and it would not time out if no one answered the call. I didn't take it apart to look for a time adjust pot (and how would it know if the call had been answered or not, it would just disconnect after the time delay, whether it was still ringing or connected). Since I had it hotline to a group, I turned on Voicemail for the group and put a Leave action in VM connected to a disconnect.

Also on IPO in Essential or Preferred, set the single line port to IVR so that it sends a positive disconnect signal to the door box, and it doesn't go blaring dial tone when you hang up.

For the OP, I would disconnect the push button and wire it to an actual doorbell. You can still call the door box with auto answer if you decide you want to talk to the visitor instead of just blindly buzzing them in.