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I think it should say "if there is $d\subseteq \omega$ such that for all $e\in E$, the intersections $e\cap d$ and $e\cap (\omega\setminus d)$ are both non-empty"
If every finite subgraph of $G$ satisfies K\"onig's Property, then $G$ has no odd cycles and is thus bipartite. Aharoni (K\"onig's Duality Theorem For Infinite Bipartite Graphs) proved that if $G$ is bipartite, then $G$ satisfies K\"onig's property.
Related to my comment above. One way to decide if a graph can be constructed in this way is to first check whether the graph can be embedded into a k-regular multigraph. If not, then it cannot. If so, then are any of the k-regular multigraphs it can be embedded into 1-factorable?
@BrendanMcKay True. But I meant it in the sense that if the OP views it as a multigraph, it may help lead to some relevant literature. Basically, given a multigraph you can decide to only view the underlying graph, but not the other way around; so there may be some benefit to temporarily thinking of it as a multigraph.
Another trivial thing is that these are 1-factorable multigraphs (that is multigraphs which can be decomposed into perfect matchings), hence regular multigraphs. A google search for 1-factorable multigraphs will turn up some papers using this language. Note that not every regular multigraph is 1-factorable though.
This probably doesn't answer your question, but there is a discussion of Reimer's proof in Section 6.4 of "The Journey of the Union-Closed Sets Conjecture" by H. Bruhn and O. Schaudt which may have some pointers to something useful.
