Sorry, I kinda forgot about this over the holidays.
This morning I soldered one of my EC2-3TNU relays to a small protoboard an added header pins to make it breadboard-able so I could test this out. Here are my results:
1) I could not make the Bean operate the relay reliably with either the on-Bean coin cell or two AA batteries. The coin cell really didn't work reliably at all. With the AA batteries even though I could trigger the relay manually by connecting VCC directly to the relay coils, I could not get the digital IO pins to trigger the relay either directly or using 2n3904 transistors as switches.
2) Using a voltage regulator (AMS1117-3.3 module) to convert a 5V power supply to 3.3V and then powering my circuit with the 3.3V I was able to get the Bean to trigger the relay reliably, but only if I used 2N3904 transisitors as switches -- not with the IO pins directly to the relay coils.
3) Using a 3.7V LiPo rechargeable battery to power the circuit I was able to get the Bean to reliably trigger the relay both using the 2N3904 transistors as switches or without the transistors and just triggering the relay coils directly from the IO pins.
These results really surprised me as the relay is supposed to work with a minimum 2.25V on the coils. The 2 AA batteries (fresh) should have been plenty of voltage to power my test circuit. I may redo my tests later in case I happened to grab a bad relay or if my breadboard connections were introducing too much resistance to the circuit. I will report back if I get better results.