I decided to try a Varta battery based on this thread. Did not make a difference.
The Cressi rep I've been emailing with is insistent that I work with a local retailer to get it fixed, and pointed me toward Eco Dive Center. I called Eco and they confirmed that while they don't repair computers in-house, they could send it to Cressi and get it fixed for about $40 in labor, plus the cost of any necessary parts, which they estimated at somewhere around $50 depending on a host of as-yet-unknowable factors.
I don't think I'm going to do it. For ~$90 I could buy a used computer, and I wasn't that fond of this one anyway. Maybe I'll see it anyone wants it for free, knowing what it'll cost to fix it, just to keep it out of a landfill.