110V shore power cannot run AC and charge the Pebble together

To follow up on the issue of the 110V plug not working when the AC is on, I got this response from Pebble:

Regarding the 110V outlet and air conditioning question, a standard 110V household outlet just doesn't have enough power capacity to run the air conditioner and charge the Pebble's battery at the same time. The AC alone uses a good amount of that outlet's available power, and the charger needs more power than what's left over, so trying to run both together would trip the breaker. This is why a higher-capacity circuit, like the 30 amp setup you had installed, is needed to do both at once. I know this created a real hardship for you with the electrician cost, and I'm sorry for that inconvenience.

Yet another thing they don't disclose until you run into it. I'm not an electrician, but I don't understand why the charging can't go straight to the battery and the battery, which has plenty of current, could be used to run the air conditioning.