The quiver given in the question has five simple modules, six which correspond to a single arrow, and the remaining representations have support as follows: 123, 124, 125, 235, 345, 1235, 12235 (note the dimension over vertex 2 is 2 in this case), 2345, 12345 (note that the 42 arrow acts as zero in the last two).
Note that the 1235 subquiver is an acyclic subquiver of type $D_4$, so it has the 12 representations which Gabriel's theorem guarantees.
I think it is clear that there is an indecomposable representation corresponding to each of the dimension vectors I listed (though ask me if that's not clear). It isn't obvious that I have successfully listed all of them, but we know there must be as many as the number of non-initial cluster variables for $D_5$, so that justifies that there shouldn't be any others.