Inverters are too much, especially for this. They dont play nice with battery power.
A lower voltage LED is the correct solution. To be honest, the 3.5Vf LED would still work as the forward voltage is only the voltage for the maximum curent draw. If you cap the voltage going to the LED (like using 3V battreies on a 3.5V LED) then the current will be limited to the voltage and the LED will still work, just at a lower brightness than what it is capable of.
Of course the problem lies in the fact that the batteries waer down and the voltage will get to 2.4V before the controller will start to complain about it. But once you get lower than 2.8V, the LED becomes too dim, i guess.
Anyway, the lower Vf LEDs are the correct solution.