Using a relay to activate a button

So, I'm trying to create a remote dog treat dispenser. I decided to start with the Manners Minder remote treat dispenser because it comes with a small remote control that dispenses a treat when a button gets pushed. After browsing around the internet, it seemed to me that the same arduino setup used in the project should work. It seemed to me that so long as the remote has two contacts that I could solder to a relay, I should be able to control the remote using the arduino. Once I opened up the remote, however, I discovered that the buttons look identical to the reset buttons on the arduino uno, and have 5 prongs instead of four. I assume the 5th contact serves as a ground, but I'm not sure.

As a result, I'm now somewhat confused as to how to wire this to the relay. Do I need to connect the ground to the relay board? Does it matter which wire goes from the NO and/or COM contact to either side of the button?

Any advice would be greatly appreciated. I can post pictures if necessary also. Thanks!

I can post pictures if necessary also

Yes that would help. Find what contacts are made when the button is pressed by probing about with a meter. If you haven't got one then get one, less than $10 from a thrift shop.

Picture attached. The button I’m trying to operate is the one on the bottom.

I don’t have a meter, so I’ll have to get one or find someone that has one.

Thanks for the picture. It looks like it could be a change over switch but you will need a meter to find out.

Is it an infrared remote control? If so, you could have the Arduino generate the correct infrared command instead.

I believe it's an RFID remote.

Bumping this with another thought/question:

If I can bypass the button by connecting the contacts to each other (which I know will close the circuit, since it'll dispense a treat when the battery for the remote is installed), can I wire the relay to the + and - terminals on the remote for the battery? I assume it'll matter that the remote's battery is 12v, and the arduino/relay is 9v. Is there a way around this?