While the question appears to be one of the legal or licensing problems with length, the real question here should be of overloading the chassis on the coach. Brakes, radiator and suspension limits need to be addressed.
The three most popular Cummins engines are the ISB, the ISC/ISL, and the ISM. You'll need at least the ISC/ISL and the ISM would be better. Unfortunately, the ISM is available on only a few top end coaches. The ISC or ISL with the side radiator on a chassis with full air suspension and air brakes would be a good choice.
As for using a gas powered coach, I wouldn't recommend it.
As for length limitations, I've never heard of anyone in a motorhome getting stopped or checked. You won't need another license if the license you have in your state is sufficient.
My wife hasn't used the inverter to power up her hair dryer, but we have used the microwave. Both use about the same amount of amps. The microwave is only on for a minute or less, generally 15 seconds or so. The micro was wired that way when the coach was built and I never changed it. But, we use the micro very sparingly on inverter power.
Hair dryers can take up to 1,500 watts, or about 12 amps at 120 volts. That is a lot of power. Now, if you have four (or more) house batteries and an inverter that can provide at least 15 amps at 120 volts, you'll have enough for a few minutes of hair drying time. Your wife is probably wanting to use it for several minutes or more, which could really draw down your batteries. But, then you'll have to recharge the batteries anyway, so why not just run the dryer off of the generator?
Good luck. (And, and you indicated, happy wife equals happy life.)