What voltage are you seeing into the board (Vin) and out of the regulator (Vcc)?
I would run the 6V thru a single diode to drop some voltage and connect to the Vcc pin. You don't need a lot of regulation, just a little voltage drop to get away from the Absolute Max voltage rating of 6V.
A standard diode, like a 1N4001, would work but it will have a constant voltage drop regardless of battery voltage. A 5.1 V zener diode will give a stable "ceiling" value of approximately 5.1 VDC
. However, the cheapest and most readily available, like the 1N4733A, won't be rated to the likely continous current draw. Therefore you'd either have to find one with a higher continous current rating, or put a few 1N4733As in parallel. If I were you I'd choose the former rather than the latter, because the electronics will be in close proximity to your head (these low voltages won't do much harm, but in some circumstances can still cause unpleasant shocks).