How long is a ‘day of continual use’? How many hours?
eneloop AA battery capacity is about 1800mAh, (going from memory) so 18 hours to run the batteries flat would mean about 100 mA of continuous current draw. 9 hours until dead would be about 200mA load current.
3 eneloop AA batteries in series will deliver about 3.9 volts. Feeding a full 5 volts from a strong USB charger could over current the LEDs on the lighting string. If the LED string is actually rated for use with three fresh alkaline AA batteries (circa 4.8 volts) then powering it with a direct feed from USB 5 volts could work nicely.
Just wire the +5 volts from USB cord to + terminal in battery housing, and ‘black/ground’ wire from USB charger cable to - battery case contact.
Somewhere I have seen ‘dummy’ AA sized battery ‘shells’ for sale, with wire leads to do exactly this, power the device externally.
The two strings that utilize only two AA eneloop batteries may also work out nicely. One string battery contacts would be connected in series to the other, then the paired strings powered from a single 5 volt USB power pack.
Commercial products like this are sold as Battery eliminators