Alright, having figured out some logistics here, I'd like to present some options to the group. As I see it, there's four different ways we can structure this.
1)
"Round Robin"-Style
Every song faces every other song over the course of the chat party, with the panel voting for one or the other, and ties broken by the chatroom. Songs are assigned 3 pts for a win, 1 pt for a loss. The song with the most points at the end is declared the winner. This method has the potential to be a little long and drawn-out--I've counted, and it will require a total of 45 match-ups--and the end might be a little anti-climatic. Statistically, though, it's also the most "fair". (I have to admit it kind of hurts my head to think about it, though. I was an English/Drama major, after all.)
2)
"Playoff"-Style
We divide the 10 songs into 2 "conferences" of 5, then do a quick vote to determine the songs' ranking within their conference. The bottom songs from each conference are dropped, and the remaining eight are bracketed like so.
"East".............."West"
1
............................1
..> A
..................B<
.
2
............................2
.........> X vs Y <
.......
3
............................3
..> C
...................D<
.
4
............................4
Again, ties go to the chatroom. This method only requires between nine and thirteen votes total (depending on how difficult it is to rank the individual songs within conferences), so it allows more room for discussion and debate, and is a little more climatic...but it's kind of bogus in the sense that every song doesn't get to face off against every other song. Also, for the purposes of the podcast, I'm concerned that we might blow through this too quickly.
3)
"Survivor"-Style
Exactly how it sounds: in this method, instead of voting for a winner, in each round the panel is instead chosing a song to "vote off the island", until we come down to the final two. Then we vote for the winner. This method is by far the easiest to manage, is probably a little more "fair" than Playoff-style, and has the potential for some pretty heated and exciting debates...but again, it's hardly scientific, and we might blow through it too fast.
4)
Some weird Frankenstien mashup of the above 
We could easily mix and match elements of any of the three styles--say, divide the songs into conferences, then do a round-robin within each conference to decide the final match-up, for example--in order to draw things out or make it more interesting. Whatever you guys want.
Of course, if anyone has any better ideas, I'm all ears.
Let me know what you think, guys!