The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). We need asymptotic conditions (or boundary conditions, there is a way to pass between the two) otherwise the kernel of our operator is infinite-dimensional. Look up the Atiyah–Patodi–Singer spectral boundary-value problem. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's "Monopoles and 3-manifolds", specifically Chapter 17.
See my comment (my lack of confidence) about answering your question for $p>2$ and $X$ compact. There is a trick I found in Hormander's "The analysis of linear partial differential operators III", Corollary B.1.6, that gets you what you want when you already have your linear elliptic operator on $W^{k,2}$ (and hence the Fredholmness for all $k$): If your continuous linear operator $W^{k,\,2}\to W^{k,\,2}$ restricts to a continuous linear operator $W^{k',\,2}\to W^{k',\,2}$ for $k'>k$ then it also restricts to a continuous linear operator $W^{k'',\,p}\to W^{k'',\,p}$ for $k'>k''>k$ and any $p$. <--- Not quite, I am not sure the "Besov spaces" Hormander uses are the Sobolev spaces (i.e. I don't know if the norms are equivalent).