Rick,
I suspect your diagnosis of a dead battery is correct. With a Cobalt 1 being at a minimum 6 years old, and being left to go fully dead (this is hard on lithium batteries), and with the symptoms you describe, that seems likely.
Changing out the battery is no more challenging than changing the battery in an underwater light, or a camera. With care for the O-ring it is possible for nearly anyone. And the battery compartment is sealed from the rest of the electronics, so even if there were a leak, the rest of the computer would be fine.
The issue is getting a battery to you. Or getting your computer shipped to someone who can take care of it. I'm trying to communicate with Atomic about options for non-US customers who need battery replacement. I haven't heard back, but with covid issues they are running on a partial staff. I will be contacting you about what options I can work out. With global shipping it just shouldn't be that hard.
In any case, don't throw out that Cobalt! We plan to have a very attractive upgrade for any older Cobalts, in any condition, coming up. It may turn out to be quite valuable.
Ron