D
Sorry, I have been away for a few days.
You did not answer about the input 0.47uF. Do you have it?
When you measure the 3.1V output, what is the input from the 3.7V cell?
If the drop across the LM3940 is 0.6V then that is out of spec.
I have over 10 LM3940s in operation (some SMD, some TO220). None of them behaves like that.
I can only guess yours is unstable (long wires or no 0.47uF) or the 3.7V is no longer 3.7V with the Omega2 connected or (very unlikely) the LM3940 is out of spec. I assume you are using a "real" LM3940 (ie reputable brand and not a counterfeit one). However, I doubt it would be a fake one because I cannot imagine there would be any money in making a fake version of such a cheap item.
Regds,
Dave