I think I should have enough power, as I've put an ampmeter in-line and the most I've seen pulled (when it's working using the 3v3 external supply) was 50-60 mA.Any ideas what I might be doing wrong? Something wrong with connecting the MCP1700's? Something else I'm missing?
In short, your power isn't clean enough. You can try adding some additional filtering (an electrolytic, at least 10uF and maybe more -- up to 470uF, depending on how noisy your power supply is.)