Advice on battery circuit for Raspberry Pi Laptop
All three approaches are deficient. Weak points:


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.
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.