Thank you for your quick response!
I have a model LiftMaster 3850.
On Page 13 of the owners manual it states under "SPECIAL NOTE"
Only one 398LM can be connected to each garage door opener. If additional wall controls are desired to operate the same garage door opener, it is recommended to use model 378LM wireless wall control as the secondary door control. I suspect there's some logic in the wall unit that's communicating with the controller in the lift - so it makes sense they would only want one connected.
I received my Garadget yesterday - and I followed the concept of attaching bell wires from the button on the wall controller 398LM to the Garadget. The wiring hack is working as expected.
I have some additional questions about operation of the device, and I'll look for other posts that may have already answered my questions.