Does every NO instance of this promise problem have a local refutation?
So if I understood well, your problem in the language of graph theory would be as follows:


  Is there a $k$ such that if for a $3$-uniform hypergraph for any $k$ of its edges we can select a $1$-shallow hitting set (a set of vertices that meets each of the $k$ edges in exactly one vertex), then the hypergraph is $2$-colorable?


The answer to this question is no because for any $g$ and $r$ there are $r$-uniform hypergraphs of girth $g$ with arbitrarily high chromatic number [Kostochka-Nesetril].
The girth $g$ of an $r$-uniform hypergraph is the smallest $g$ for which there are $g-1$ edges whose union has size at most $g(k-1)$.
Thus, if a hypergraph has girth $g$, then any $g-1$ edges have a $1$-shallow hitting set by induction because their union looks like a tree.
This means that locally it satisfies your 1-in-3 satisfiability condition, but it is not $2$-colorable, so there is no NAE-satisfying assignment for the whole hypergraph.