LiIon is usually charged at constant current until a max allowed voltage is reached and then is held at that voltage while current tails off under "control" of chemistry of battery until Ichg = k% of Imax where k% is chosen according to longevity or max energy concerns.50% or 25% of Imax gives longer life. 10% or 5% tail gives max capacity but lower life.
Lowering Vpedesatl by 0.1V greatly assists battery life.
Discharging to higher cutoff voltage aids cycle life.
LiIon also has calendar life and starts self destructing from day one so a lightly used battery still dies.
Best cycle life is achieved by stopping charge when Vpedestal is reached and systen changes from CC to Cv. By monitoring voltage this point can be observed. You could even do a "dumb" system that simply watched delta Vbat and declared constant V when delta fell to zero. Only slightly more than a comparator and an RC delay in one input would achieve that.(While Vin is ramping a delayed vin is lower. When Vin pedestals the delayed Vin almost catches up. An offset voltage is needed to allow comparator towork).
LiIon cells mechanically flex the cell as metallic Lithium is "plated" in and out of the cell*. Cycle life is in large part due to battery beating itself to death mechanically.(This is why LiFePO4 lasts much longer and has lower capacity - the material is held in an Olivine matrix that maintains constant shape as active material is moved in/out BUT it takes up some space. )
*Note: Bill Dubuque has suggested that this sentence would be better replaced with " 'LiIon cells mechanically flex the cell as Lithium ions are intercalated".
The distinction is a finer one than may be apparent. However, it is true to say that if you cut open a LiIon cell you would not usually find metallic Lithium in it. Bill notes that this makes primary Lithium cells, which do contain metallic Lithium, a greater fire hazard than LiIon cells.
If you charge a LiIon cell with excessive voltage metallic Lithium will be 'plated out' and "vent with flames" mode usually occurs at about the same time.
Charge to CV level as often and as soon as possible.
If charging all the way their "disconnect message" is a sign of bad ethos. They are probably trying to minimise the risk of fire without telling you.
For longest storage life (as opposed to long life in regular use) storing at a lower voltage than Vmax is in order. Probably at about 3.6 V and only about 30% state of charge. The various Mars Rovers use LiIon batteries and have a design life of about 8000 cycles - but charge to about 3.6 - 3.7 V maximum.
8000 / 365(~=) ~= 22 Terran years.
When charging lithium batteries in series, you need what's known as a balance charger to keep the cells at similar voltage. One way that some chargers do this is by stopping the charge of both when one is full, discharging the full one until they are even, and then charge them both up again. There are fully featured chargers meant for the remote control hobby, but they require setting the charge options on the screen before every charge. I cannot seem to find any "plug and play" balance charger boards similar to the TP4056. I am fairly sure that ICs exist that will handle the buck/boost power supply and the balancing, but I can't seem to find one and you would need to manufacture your own circuit board, and the chip is almost certainly surface mount.
That being said, there are a few workarounds. One is to have the batteries in series for operation and parallel for charging, and use either one or two single cell chargers. This can be accomplished through a 3PDT switch, and a good explanation is here. Note that you don't need to use the same charger, your TP4056 should be able to be substituted.
A simpler and more elegant way to do it would be to use a boost mode power supply, and don't put the cells in series. You can attach multiple of your current cells in parallel, so that if you have two cells and are drawing 5W, it would be 2.5W per cell or ~0.67A. There are many chargers that are designed for this application, handling both the CC-CV charging and the boost regulator for the output, such as the Lipo Rider Pro from Seeed Studio which will do 1A output. This is what I would recommend.
Alternatively, you can buy more powerful batteries that will put out more than 1A instead of putting multiple in parallel.
Or if you want to take all the fun out of this project and just buy something that sort of works, there are USB battery banks available from China that will take care of all the charging and discharging of 18650 LiPo batteries. I have seen banks that will take from one to six 18650 cells in parallel, and some of the larger ones can do as much as 3A output. Here are a few examples, and a video review. Their build qualities and user interfaces look pretty bad, but apparently they work.
Best Answer
I've used the MCP73831 which is a cheap linear charger from Microchip. They come in flavours appropriate to what features you want - in other words you get an -2AT if you want 4.20V, terminating charge at 20% of set current etc.
Choosing the one resistor correctly gives you the charge current. It's a reasonably narrow resistor range giving a current on a curve. From 10k to 60k or thereabouts gives you 100mA to under 20mA. Either side of that range, it's not really accurate. That range seems to be in the charge range you are looking for though.