The fan has its own lights, designed to work from two separate power switches, along with a power switch for the fan power, and the low-voltage switch you mentioned. The reason the fan has that two-switch arrangement, is that it is capable of constant low-volume operation, set by a dial on the side of the unit. The low-voltage switch is for high-speed operation, which is what you're looking for, it seems.
It seems you could skip the low-voltage switch, and use a jumper wire connecting those conductors. That leaves the fan set for full flow, whenever you apply power to it. How you deal with the lights, would be up to you, and whether you want the night light.