Along with the (defunct) Blackburn Flea, the bike pack also disgorged an anonymous taillight with a battery resistant to recharging through the USB port. Gentle suasion cracked the solvent-glued joint around the case:

As with most modern electronics, a battery occupies most of the interior volume:

For posterity, the connections:

I unsoldered the cell and charged it from a bench supply:

The voltage started out low with the current held to about 100 mA, eventually rose to 4.1 V, and stayed there while the current dropped to zero. Unlike the Blackburn cell, it appears not too much worse for the experience, although I haven’t measured the actual capacity.
Clipping the Tek current probe around the LED supply wire produced this waveform for the “dim” setting:

Adding a voltage probe across the LEDs and clicking to the “high” setting:

The intense ringing at the start of the pulse seems an artifact of the measurement setup, but ya never know; these days, RFI can come from anywhere.
In any event, the COB LED strip draws 800 mA from a fully charged battery, about 26 mA for each of the 30 LEDs. The 5% duty cycle in the “dim” setting is decently bright and 18% in “high” is entire adequate.
A trio of blinks works for daytime rides, although the fastest one seems seizure-inducing.
I’ve strapped it around a rack strut and run it at the slowest blink, on the principle you can never have too many blinky lights …