Higher voltage battery able to drive load longer at stable voltage?
What matters in not voltage, it's energy.

The energy in a battery is quoted in Wh (watt-hours). Many still don't specify the watt-hour capacity but do quote the Ah (amp-hour) capacity. With this and the battery voltage you can calculate the Wh capacity from:

$$ Wh = V \times Ah $$.

Compare the two.

If you use a step-down converter you need to factor the efficiency of that into the calculations.