Electronic – Limiting or reducing current to charge low-capacity lithium batteries

All three approaches are deficient. Weak points:

1. Manual toggle switch will fail to manage battery discharge phase, so the batteries will likely be accidentally overdischarged and die soon. The switch should be designed to work automatically.

2. Connecting two cells in parallel requires careful cell match, otherwise the charge/discharge can be unbalanced, and lead to excessive self-discharge and therefore to reduced battery capacity.

What you need is a complete PMIC - power management IC, which contains both Li-Ion charger AND "power-path management" (automatic toggle). Here is an example, from SparkFun.

If a single 18650 cell does not have enough capacity to run your device, use two or three cells, but in-series, not in parallel. In this case you need to use PMICs designed for balanced charge of multiple Li-Ion cells. Look up Digi-key or Mouser for 2-3 cell PMIC/chargers, pick the price point/brand you like, then go to manufacturer website and get application notes.

And yes, if you use a right PMIC and if the delta between supply current capability and battery charge current (which is set by resistors) is enough to run your device, the PMIC should be able to charge the battery and feed your device. But you should check with IC specifications.

The residual energy in the AA cells is not valuable, the fact that the cells are portable, safe and cheap is.

That means using them for battery charging, that could use wall power, which is a few orders of magnitude more powerful and several orders cheaper (than full price batteries at least), is a total inversion of values and waste of time.

I too had access to a stream of partially used discarded AAs, and was able to make a steady donation of them to a charity serving the visually impaired, for use in their handheld illuminated magnifiers. They were perfectly matched to this service, as premature exhaustion of the cell was of no consequence to the users, especially given that they were free. The key here was a reliable stream, which allowed the service to abstain from buying full price cells.

If you insist on trying to recover the energy (I'm trying to imagine a desert island or post-apocalyptic situation where you don't have mains power, but do have the theatre's discarded AAs), then a) mixing AAs of a different charge state will result in premature run-down of the string of cells and loss of efficiency rather than damage (as long as you remove flat cells promptly) b) inverting up from a single cell is the least efficient way to do it, better to use a higher voltage and c) 18650 can be charged by a low current without reduction in life, as long as you continue to observe the normal charging envelope.