@pjf@cloudisland.nz
I'm too impatient to wait for a larger capacity battery, so I'm disassembling an old laptop battery to see if I can find any usable cells, and if I can scrounge a lipo protection circuit from somewhere to go with said recovered cell.
@pjf@cloudisland.nz
Aha, looks like I need Maximum Power Point Tracking (MPPT) doodad for charging off a solar cell, and I doubt my $10 panel already has one integrated.
If I understand correctly, an MPPT doodad lowers the current draw from the panel to maintain enough voltage to charge; otherwise the battery charger can try to draw too much, which drops the voltage, and one gets nowhere.
A random blog post recommends a CN3791, which are only a few dollars, so I shall continue investigating