Monitor the voltage inside the RV when the air conditioner is running. The manufacturers say the lowest allowable voltage is 115 minus 10%, or 103.5 volts. Here's the info from
http://www.airxcel.com/coleman-mach/service-support/faqs#:
What is the proper operating voltage for a domestic RV air conditioner?
Proper operating voltage is 115 VAC/60Hz. The unit may be run safely between 104 volts and 125 volts. Do not operate the air conditioner outside of these parameters, as serious component damage may result.
I'd give it a couple volts padding, and make sure the meter is reasonably accurate. As far as blowing a 15 amp breaker because of the starting current, that's hogwash. A 15 amp breaker will carry a minimum of 75 amps for 1 second, 41 amps for 5 seconds, and 30 amps for 10 seconds. The trip characteristics are specified by UL 489. The previous advice to check if the cords or plugs get hot is all good.