I've already written this as a comment on Jan's project about his blown-up LDO, about how I've been struggling with a similar problem, but I wanted to post a project log update as well.
During recent, normal operation, I had the pleasure of hearing a sudden "pop" somewhere inside the casing. Now, as far as I know, that's not part of what these chips are supposed to do, and when you smell magic smoke, something is definitely broken.
In order to prevent even more from breaking, I had to take the casing apart as quickly as possible, and cut the wire of the LiPo, as parts of the charge circuit were connected directly to the battery, with no switch in between.
During "post-op", I managed to get a closer look at what had happened. There are no visual marks or popped ICs, but I am very sure the charge IC, a MCP73832T, is at fault, because the "charging" LED had been glowing with no cable plugged in. In fact, this is the same chip that failed on my Lasertag project!
Although I might not have added aaaaalll the smoothing capacitors, there was a 10uF on the input and output side, so I am definitely going to look for a beefier IC.
As for the DSKorder itself, I was able to desolder the broken IC and power the circuit via USB. Nothing else was damaged, and even the USB-To-UART chip, which is powered from the same line as the Charge IC, responded just fine.
I'll replace the broken IC with an external 500mA charge board, so ... All's well!
Just need a better charge IC for next time.