DIY Use 18650 in place of 18500
The limiting factor may be the Solar panel and not the battery.  5Voc to 3.7V @ 30mAh in max sun.

So depending on LED load it may be 1:1 or 2:1 charge:discharge ratio per day full direct sun hours full sun:dark hours

If you live in a very sunny climate, it may help a bit but may never fully charge with double the battery capacity. If you get 4 good sun-hrs around noon and only get 1 hr of decent light, a better battery may help get 2h. 

It's just a SWAG.

3x5" cells can be arranged in series or parallel to get different open circuit voltage, Voc  2.5,5,10 but the same VI power.  From a quick search, I see 3x5" panels rated for 30mA on a 3.7V cell which is near ideal at 70 to 80% of Voc.

So 30mA charge rate is max at 100kLux full sun.  Then only 3mA at 10kLux cloudy  or a wide angle  offset.

There may be 1 or 2 LEDs with series R selected for ~13mA each +/-50% depending on design then operating most of the time from 3.7 to 3.3 then dropping to 2.85V when LED off.  The LEDs are about 2.8V + If*16 Ohms=Vf then + IfRs=Vbat
Still this is just a SWAG. with 50% tolerances on any LED internal chip Rs.