Close/open a physical button circuit via Arduino
The easiest way to implement a button override function like this is to completely isolate the two systems so that there is no direct electrical connection between them. There can also be unknown things going on with how the control system inside the pet door monitors or scans the switch to be overridden so the best substitute for the switch is to use a dry contact pair. Overall this leads to using a small relay to perform the switch override.

Your goal is to get two small wires soldered into the existing circuit that connect to the two sides of the button to be overridden. These will then go to the NO and COM terminals of the relay (Normally Open and Common). The coil of the relay will want to be driven by an output pin from the MCU. Depending upon the relay selection it may very well be necessary to add a transistor driver circuit between the MCU pin and the relay coil. In any case make sure to not forget the clamp diode that is reverse biased across the relay coil to prevent the high voltage spike that occurs at coil opening time from frying your transistor or MCU pin.