First obvious question, why do you have three TP4056?
They are not designed to be connected in parallel, their outputs will affect feedback and therefore distort proper charging profile.
Second, it does not look like TP4056 is designed to be used with system load connected. You need more advanced BMS to have this functionality. The functionality is called "Power Path".
MCP73871 is an example of charge controller that can do what you need.
And here is AN1149 application note that can answer all your questions. Note that application note uses different controller (MCP73837). The MCP73871 already has most of those external components inside. This is just for the reference, to better understand the potential problems and see how they are resolved.
Using solar panel as external source presents several problems.
First, 7V is absolute maximum for MCP73871, meaning it will survive but not for long. So, you'd need 6V external voltage limiter (e.g. zener-based) for the times when solar panel output is too high.
Second, very often the panel output will be less than 4.5V required for charging, meaning your battery will only be charged in bright light.
Third, peak available power means very little for CC/CV charging profile of the LiPo battery. You either using very little of it, or you can use much more than 2A, depending on the current stage of charging.
The only way I see to solve all these problems is to use MPPT like SM72441 between the panel and BMS.
- The MPPT can be programmed for fixed 6V output. It will boost the voltage in low light and limit it in bright light with about 99% efficiency.
- It will keep solar panel at maximum power point. You won't be using "peak available power", you will be using maximum available power at all times.
You can buy ready-made MPPT modules for just a few bucks (no guarantee of quality though) or design your own.