Charging 3 Li-ion batteries in parallel and discharging in series
No offense, but what you propose is a kludge solution. There is a strong possibility of a catastrophic failure due to relay timing issues and failing to do proper charge management.

There are battery charging ICs that can charge these cells in series, take care of charge balancing, and no need to do any switching to draw current from the battery. As one example of a very basic chip, take a look at the LTC4079. There are many such chips on the market.

Here is a simplified circuit for how to use a chip like the LTC4079:







simulate this circuit &ndash; Schematic created using CircuitLab

Note that the LTC4079 does not include cell balancing. Most cell balancing ICs require microprocessor control, at least to set it up initially. Take a look at the ISL94203 chip as an example of one that you can configure after which it will run autonomously.