Update: This is very bizarre.
While in the game, I navigated through some menu windows and found a bookmarked map. For some reason, SC2 had no problem downloading this map and playing it. After a game against the AI, I returned to the main game menu to find all features working properly. (I haven’t tried multiplayer yet, but I am able to get maps and play games in the normal fashion.)
I have no idea why this worked. It is possible that playing a game simply left me in the game long enough for connections to the server to get going, or downloading a map got the game working. (I’ve left the game running for 5 minutes before to see if it would help, but the game vs. AI was 10 or 15 minutes.)
At any rate, the game is working (persistently, I checked). If anyone has a similar issue, I recommend finding any map you can play on to get the game to download something.
Thank you for the replies LeJimster, I’ve been using Linux for a couple years now, but this is my first foray into gaming on Linux. I’m glad to see such an active and capable community behind it.