TouchToneTommy

The Partner phones have an "AUX" jack on the bottom of them. This will drive a Tip/Ring device. Avaya (actually Plantronics) makes a "Partner Base" which is actually a single line phone in a box, with an On/Off button and a headset connector.

You turn it ON, get dial tone through the headset, but can dial and operator all the buttons on the set to handle your call. Turn it OFF when finished with the call. Never touch the handset.

If your only option is a traditional headset that connects through the handset connector, and has a "headset/handset" a/b switch, you can sort of cheat. Assign an unused line to the set and label it "hang up". When you are using the headset, take the handset off the phone and leave it off. When not on a call, press the line labeled "hang-up". You are actually off hook on a line without dial tone. Press a line when you want to talk. Go back to the hang-up line when you are done. Put the handset in the cradle when you are done with the phone altogether.