I created a brand new character with a different Shared SNAK than I used before (found one online), and the game appears to allow me to save properly now (???). Sorry to bother you all.
Follow-up question: There's a lot of differing information out on the web about SNAK and sylverant server. For dreamcast PSO Ver. 2 NTSC-U, if I backup my save files across several VMUs regularly, are there really any additional advantage (server events? How much should I bother?) to having a private SNAK? Copies of this game are ridiculously priced on eBay now.