Upsizing Li-Ion battery
The speaker almost certainly uses a LiIon battery.
It may have more than a single cell, but usually not.
Measure battery voltage when fully charged and when the voice advises.
A single LiIon cell will have a fully charged voltage of close to 4.2V and a fully discharged voltage or around 3V - but the latter may be lower or higher depending on design.
Multiple cells will have multiples of these voltages.
Assuming a single cell, you can substitute a higher capacity cell -  say an 18650 with typically around 3000 mAh capacity, with a very good chance of success. It is possible but very unlikely that the original charge rate is higher than the 18650 will tolerate - but in almost all cases small LiPo cells will have charge rates in the C/2 - C range. (C = 1 mA per mAh capacity).
Substituting a higher capacity cell will have minimal effect at the low voltage end.
When fully charged the charger will usually terminate at some fraction of the usual capacity - maybe C/4 or 250 mA for a 1000 mAh cell. When charging a higher capacity cell the terminate current will be a smaller fraction of C - eg for 250 mA terminate current and 3000 mAh capacity this will be 250/3000 of C or C/12.
This will result in the cell receiving a higher % charge - leading to a lower overall cycle life - which is unlikely to be a major problem in this case.
If you wanted a &quot;charged&quot; indicator you could add a monitor that detected when Vbattery reached 4.2V. This occurs at the start of the CV portion of the CCCV charge cycle. Stopping charge then will reduce capacity stored by perhaps 10% but will greatly increase battery cycle life.