A partial answer: a result of Juraj Bosák says that if all vertices of a graceful graph have even degree, then the graph has $4k$ or $4k+3$ edges for some integer $k$.
Since $K_5^{(3)}$ has $30$ edges, it is ungraceful.
Gallian's survey attributes this result to Rosa's 1967 paper On certain valuations of the vertices of a graph. See also Don Knuth's work-in-progress fascicle 7a (https://www-cs-faculty.stanford.edu/~knuth/fasc7a.ps.gz) for discussion of computational aspects of graceful colorings as constraint satisfaction problems.