Quick update: I tried the circuit, relays connected to power (with the diode bridge) and NPN collector, signal to base with resistance and emitter to ground. All looks good, although now I need to correct the power for handling the relays safely between voltages (let's say, 4V to 9V) as Tom said.
However, and just for the sake of completeness, the rebooting problem still stands when using the 3.3 out from the moteino. 4 half depleted AAA batteries (around 4.5 V) handled it fine (although couldn't power the two relays at the same time for more than 5 seconds), a 9V battery reboots, and the Li-Ion battery keeps working. Mark, you are probably right.
At this point I am ready to write off using the 3.3 out, it was a silly idea. I now understand the 3.3 out is there just to output voltage into buttons, switches and leds.
Since this post was moved to projects, I will keep posting followups about it. Thanks for the help so far, really! Further feedback will be greatly appreciated.
I need to ask my dad what kind of current will go through the relays. Depending on that, maybe I can switch to solid state relays or even optocouplers (? please correct me if an optocoupler is not suit for switching low power independent circuits).