The only reference that I can find to an "Appleby House" is in Richard Dunmore's analysis of the 1841 census, in which he used a parish map from 1832.
This shows an Appleby House which was much enlarged and extended to create Appleby Hall, as Richard mentions that conversion took place in the 1830's.
But I don't think the old Appleby House was ever a roadside inn as it was the home of the Moore family.
There is a pub in the village on the main Atherstone Road (A444) which is now called the Appleby Inn, this was previously known as The Moores Arms, but before that it appears to have been occupied by a local farmer.
This is the link to Richard's article if you haven't already read it.