An update.
As has been suspected, the ~.6V drop in the Joye products is built right into the FET switch circuit. It is not due to internal resistance in the battery itself.
I have pulled apart one of my Mega batteries and am doing a circuit analysis. Although the analysis isn't anywhere near complete, what I can tell you is this:
With a 2.5Ω load applied and the switch energized, the measured voltage right at the battery is approximately 4V (freshly charged). This is also the FET source voltage.
The drain voltage is in the 3.1V range (as previously noted) and this is the output voltage of the complete package.
The gate voltage is approximately .8V.
I haven't gotten far enough to figure out where the gate voltage is sourced from...... but I will!
NOW were talkin! I'm excited to see if we can get the intended voltage out of genuine Joye batteries...