Li-Ion Battery Solar powered circuit
Measure your load and charge current accurately. 


The battery storage efficiency depends on heat loss and impedance matching or MPT matching V=72~82% of Voc.(open cct) Otherwise, PV efficiency is poor.


Now your charger is demanding too much current from PV since you can't get 1A from a 1W 6V PV ( photovoltaic array)  AN optimal charger is not used, so you might only get &lt;50% of rated ideal power out.  The Programmable charge current must not exceed the supply or else the voltage drops and thus power available is less than ideal.  When a battery is 3V and PV has 6V your charger drops the PV to 50% voltage while charging thus cannot deliver 1W.
The charger is only "smart" when the supply is greater than the load of 1A.  Otherwise, it is slow and poor efficiency. 
I assume 6V PV is the rated voltage for a 6V battery @ 1W thus 160mA max only in full direct sun.    In theory, your load is about 5.2mA average. 250mA*2s/120=4.17mA +1mA = 5.2mA  . SHow actual product specs for everything.
So if your PV charges say 180mA 10% of the time and 60mA 33% of the day and low current otherwise, that's an average of 18 mA and 20mA or 19 mA average. so it may be enough if the storage efficiency in the battery is 50%.  But it is marginal and that was hypothetical. 


idea

You could use a current sensor and Op Amp to make a V to f converter with a counter to accumulate charges as binary counts then do the same for current discharge using LED\s to display the count or more sophisticated Coulomb counters.

 reality check

But something says your current load exceeds the storage from the PV or the battery capacity is less than you expect.  Frankly, a 6300mA Li-Ion sounds about 200~300% over-inflated and may be fake specs.