Has the OP considered that the TP4056 can supply current for the project as well as battery charge current?
Yes the TP is limited to 1A and sipping power from charging the battery will cause the battery to charge longer but both battery charge and load supply can be had from just the TP. I did such for a while with a solar project using a TP4056.
Could we get some information on which batteries you will be using, and the voltage of the charging power source? It just seems that relays draw a lot of current at least half the time, and perhaps there's a better way of providing battery power that wouldn't require a relay at all.
using a bi-stable relay that needs just a 50 milliseconds pulse to switch to the other state
or using MOS-FETs to switch between the two batteries.
If one battery is always loading there must be somewhere a permanent non-battery-power-supply.
So why use batteries at all?
Of course depending on your application batteries might be required. But you should explain why you think you need two batteries