I chose the "DW-CT14" module, found all over eBay and Amazon. It's got a Bluetooth 4.2 chip, two amplifiers and circuitry for charging a 3.7 Li-ion cell (as long as it has protection).

The amplifiers (NS4165) are capable of more than 5W into 2ohm, but into 4ohm can only manage 3W (from a 5V supply) or a measly 1.8W from the 3.6V I'll have from my single Li-ion cell. So those listings that advertise the module as "5W+5W" are a little way off! The chip can be switched between class AB and class D and, thankfully, they do appear to be configured for class D. Earlier versions of the module seem to use the TC8871 instead, which is a pin-equivalent part.

The BT module is in a 24-pin SOP packaged marked "AS19AP" which seems to make it one of the ubiquitous JieLi ("Jerry") family, but it's not clear which. It's not an AC9605, judging by the pinout which is different. The USB micro B port powers it (and charges the cell), and its data lines are connected to the chip, but no device appears when connected to a PC, so perhaps it needs putting into a special mode (as described here for other similar parts). There are three test pads connected to pins 13, 14 and 15 which I suspect are CLK, MOSI and MISO - might be fun to try connecting at some point.

The whole module draws 14mA when paired but not playing, and 41mA when playing at minimum volume. 30mA of this would appear to be the two amplifiers if the datasheet is to be believed. From my 1400mAh cell I should easily get a day's play time unless I run it quite loud. Four days if I don't play any audio (i.e. just BT).

Li-ion charging current is approx. 125mA (measured with a nearly-full cell).

As a first test run, I 3D-printed a partial enclosure, Blu-tacked the speakers in and wired everything up. Despite the puny output power, the whole thing does seem to be reasonably loud - enough for a picnic but not a party. The worst part is the awful Chingrish announcements when switching on. Surely a short beep would have sufficed?

I'll probably want to switch the module off to conserve battery. The obvious way to do this would be to interrupt the connection to the battery, but this means it can't charge unless it's switched on. I might see if I can bodge a way to put the BT and amplifiers into standby, rather than powering-off the whole circuit.