I would hazard a guess that there are no IC boost regs that can hold 80% efficiency boosting 2v (0.6A) up to 3.7V @ 350 mA. The peak currents on the 2V supply would go through the switches. Like I said, it's theoretically possible but I think the efficiency would be pretty bad. 60% would actually be pretty good for this. I believe Maxim has the biggest selection of low voltage boost ICs.A good DC-DC boost converter will be 80% or better efficient.
So a bridge rectifier will get you 2.8 Volts DC and you will still need a low voltage DC-DC converter with all the same problems mentioned by bountyhunter. Where is the cockamamie requiremnt coming from?i forgot to mention,the input is 2V ac