This circuit alone is only meant as a battery charger, not as a complete load sharing solution.
As you got correctly (and as is the case with any charger IC that doesn't handle a specific load path), if you connect a load in parallel with the battery, the charger IC has no means of knowing when the charge is over, as it relies on the current falling below 1/10 of the charging current when the charger is in constant voltage mode (last part of the charging cycle.)
Some charger ICs have an additional internal time-out counter, meaning that if the current has not fallen below 1/10 of nominal charging current after a given amount of time, the charging will just stop. I don't know whether the TP4056 has this feature. If it does, it would just shut down the load automatically after a while if said load exceeds the threshold current.
Another issue with this setup is that if your load is not a constant current load, the charging current to the battery will effectively vary over time, which is not optimal either.
Just putting the load in parallel with the battery is a bad idea overall in general and should be done only with extra care. You should preferably add some kind of switch-over circuitry to power a load while charging.