So in working out the details and the structure of the groupoids of commensurability relations on 1dQLs, I realized I could generalize the method a bit more.
Given a set S with an equivalence relation ~, we can form a groupoid consisting of all ordered pairs of equivalent elements, i.e., all pairs $(s_1,s_2)$ where $s_1 \sim s_2$. The composition is then $(s_1,s_2) \circ (s_3,s_4) = (s_1,s_4)$ defined when $s_2 = s_3$.
It's such an obvious example of a groupoid that I'm embarrassed that I missed it. I'm sure it's in a textbook somewhere. I would have saved a lot of time if I had found and read that textbook a week ago.
Oh! Look! There it is! It's on wikipedia. fml.