You got me curious again @Vitesze. I did some additional research on a chip solution to this primary/secondary, and maybe...just maybe...you could utilize the LTC4415 from Linear.
My thought is, you could use the EN1 and EN2 (enable) pins on the IC, controlled by digital outputs on the Electron, to switch to the Primary (non-rechargeable) battery when the Ublox module is transmitting, and then switch to the Secondary (rechargeable) battery for the rest of the time when the Electron is finishing up code and sleeping / deep-sleeping.
It's not a perfect solution because it's not actually detecting the current from the load and switching to the primary battery, but maybe it can be used in conjunction with that.
The output of the LTC4415 would go to Li+.
Unknown: I haven't read the whole datasheet but don't know if the switching between batteries has a delay that would shutdown the Electron because it's not receiving power.