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The data also shows the regulator board I took out of a drawer is an LDO, not a buck converter, so it's 60% efficient a lot of the time. I will make a pin compatible replacement with the converter I specified for the final boards and that should be a big help.
I've now set up two 1W panels in parallel to make an effective 2W panel. Only a few hours later it's clear this makes a massive difference as they spend lots of time charging the cell rather than just 'treading water'. I'll leave the test to run until the battery protection kicks in at 3.5V but it looks like 2W panels really are what's needed even if I were to replace the LDO. There are some really quite affordable 2W 5.5V cells on Banggood, so going up in size isn't a big deal.