It looks like the power problem is now sorted following the fix I mentioned in my last log. I'm now getting about a month per charge, and if I dropped the low-battery threshold to 3.2V I could probably go two. Better than I'd hoped!
I'm now working on the Mk2 which will use a custom PCB and no off-the-shelf modules. It will feature an actual battery cut-off when the low-power threshold is reached so the LiPo will not be damaged. The idea is that a FET is in-line with the battery. When plugged into USB, the FET is held on. The micro-controller can also drive the FET, and will hold it on as long as it detects the battery is OK. When the battery drops below the threshold the micro-controller will allow the FET to turn off. That of course also kills power to the micro-controller, so the only way of waking things up again is to plug it in to charge. Should work.
I have the bare OLED displays via EBay (i.e. not on a breakout/carrier board) and will need to now add the supporting circuitry myself. I think with all this I should get the thickness down under 10mm (display + PCB + battery). I'm currently trying to use bits from the seedstudio OPL so I can use their fusion service, although soldering the SMD stuff is no longer as intimidating as it was.
I'm considering making these generally available (with the addition of some breakout points for unused pins) as a small "wearable" board with OLED, battery, charging circuit and micro-controller seems it may be of interest to others, and this is a different form factor to most I've seen which seem to favour larger colour and/or squarish displays.