I suspect that one reason RV parks say NO is that there is information that says NEC says no.
Here is a link
http://www.codebookcity.com/codearticles/nec/necarticle551-1.htm
Note the line You cannot use autotransformers with RVs 551.20(E).
so it may have nothing to do with "power stealing" at all.
Whats not clear is if this applies to the park side or the RV side.
Also, one thing to be aware of with an autotransformer is there is NO isolation provided by the transformer. the input and output sides share a common neutral, unlike a more conventional transfomer with primary and secondarys isolated.