How to close a dry switch with an Arduino?

I’d like to let the Arduino control a VGA switch. It’s a Kramer VP-211k that has connectors for a remote (dry switch). I’ve meassured it to about .01V-.03V. I have used the device with a flip switch (on-off-on) to close the circuit and it works perfectly. But how do I replace the flip switch with an Arduino? See attached image.

It’s way to low current for a relay, right?

The MOSFETs I’ve been looking at go as low as 5V but not lower.

Or some other kind of transistor?

I think I’m missing something major conceptually. What component do I use here to close the circuit?

It's way to low current for a relay, right?

No. Any relay would work but a reed relay is a good fit.

The MOSFETs I've been looking at go as low as 5V but not lower.

That doesn't make sense. What will not go lower than 5V? You are misunderstanding what you are reading.