The heat from the combustion of Brown's gas is not going to improve mileage.
You're taking mechanical power, using it to generate electricity (at less than 100% efficiency), using that to split water (at less than 100% efficiency), then burning the resulting gas back to water releasing the remainder of your investment as heat - on the hot side of a heat engine limited by the carnot cycle efficiency. That last is really bad news efficiency-wise.
If this worked out you'd have a perpetual motion machine of the first kind (over-unity). Then who would need gasoline at all?
Which is not to say it's guaranteed NOT to improve mileage.
- A small admixture of hydrogen and oxygen might improve the propagation of the flame front and result in better combustion of the regular fuel.
- The generator might result in the injection of small water droplets, which convert to steam and improve the engine efficiency by better coupling the heat to pressure pushing the piston during the power cycle.
Either or both of these might end up improving the engine efficiency enough to more than offset the power spent generating electricity to generate gas.
I know that one of those (water injection) has been shown to improve the efficiency of otto-cycle engines dramatically. (The auto industry doesn't use it, much as it would like the efficiency improvement, because of the problems from requiring TWO consumable liquids to "fuel" the vehicle, and what to do when one of them runs out.)
Speaking of which: I understand one "magic carburator" that actually works pretty well is a gasoline-water-ultrasound device. It emits a fine coaxial stream of water surrounded by gasoline with ultrasound waves inside it. This breaks it into a spray of tiny, standard-sized, droplets of gasoline-coated water. When these burn they ignite like a rich mixture (because of the outer gasoline coating's increased surface area). Then the water boils, blasts the remaining unburned gasoline through the cylinder, and provides added steam pressure. Combustion completes like a lean mixture, but with the water cooling the gas below the nitrogen-fixation temperature and retarding knock. Low emissions, increase in effective octane, more complete combustion, water-injection "steam power" assist. All around-great. Until you let the water tank run dry while you still have gasoline...