I haven't worked with the Arduino. Would it be difficult to incorporate an SPDT relay to switch the sound source between it and the modem?

Hmm, probably not, actually, that's a good idea. It was one of my first projects, so I didn't really know what I was doing, which limited how much I could do.

It's a great project! Thanks for sharing it.

If the relay works you could also simulate the pulse dialing signals with clicking audio on the dial return.