Dock melted a spot on cover and smells like burnt. Won't hold charge

Bad answer from Apple. Get some good looking young lady to take it to Apple and get them to replace the unit.

Sounds like your dock cooked the battery.

The high thermal energy from the dock most likely heated out the ions in the electrolyte of the battery, no ions, no charge.

Only option is to replace the battery.

since the replaced battery still dies quickly, the logic board - or the dock connector - has one or more parts shorted together/toasted/bbq'ed. finding that part shouldn't be the problem - but replacing it would be pretty hard. you need someone who does component level repair every day - maybe you will find someone in your area (but i don't know if you could find those people in the yellow pages)

good luck