First, if your shore power cord is 30A then it is 30A 110V and a 15amp garage outlet will run the A/C if nothing else is on, but it is likely that your converter is drawing some AC amps too so you will soon blow the 15A breaker. You have a better chance if your garage outlet is 20A.
Second, how does your coach switch from shore power to generator power. Newer coaches have an automatic transfer switch (ATS) and if yours has one, it is probably bad. If you do not have one, then the common way coaches are built to switch to generator power is to move the power cord to the generator input socket. This is located in a hatch towards the rear of the coach usually on the driver's side.
David
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