So I built this little circuit in tinkerCAD and it does "work" but I feel like I'm not using these transistors correctly. Is there a more proper way to do what I'm trying to do here? I want the arduino to somehow create a short across the buttons using with as close to the voltage of the battery as possible.
To get as close to the voltage of the battery as possible, consider using a single-pole, double-throw relay to reroute power straight from the battery. Here's a possible schematic where the Arduino controls the relay:
Just remember to protect the Arduino the same way you would a motor (or use a relay driver board). Here is an Arduino tutorial for working with relays for reference.
Another galvanically isolated switching method is the opto-isolator. These act somewhat like a cross between a relay and a transistor. Internally they are a LED shining at a light sensitive resistor and a maybe transistor to boost the switching current (depending on the model)
These can be used in a similar manner to the relay that @NullPointerException explained.
