Yes, I was aware of the power requirements, and used a separate 3.3V supply with plenty of current. I also added pull-up resistors to the Reset, CH_EN, and GPIO-0 inputs since I couldn't find any documentation indicating whether the internal pull-ups were active.
So, I think the only issue was that I was trying to use SAP mode when I should have used station mode. After making that change it worked fine. The AT command set was very useful to get things up and running initially, but I've moved on and have made a custom application for my ESP8266 using the Arduino IDE. It's now uploaded and seems to be working fine.