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 $k$. (Proof added below.)

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 section 7.2.2.3 of The Art of Computer Programming (https://www-cs-faculty.stanford.edu/~knuth/fasc7a.ps.gz) for a discussion of computational aspects of graceful labelings as constraint satisfaction problems.

Proof. We have $\sum_{uv\in E(G)}|l(u)-l(v)|=1+2+\dots+m=\binom{m+1}{2}$ when there are $m$ edges; and a given vertex $v$ appears exactly $\deg(v)$ times in this sum. Working modulo $2$, we also have $|l(u)-l(v)|\equiv l(u)+l(v)$. Therefore $\sum_v\deg(v)l(v)\equiv\binom{m+1}{2}$. But $\binom{m+1}{2}\equiv1$ when $m=4k+1$ or $m=4k+2$. $\square$

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 $k$ (proof given below).

Gallian's survey attributes this result to [Rosa 1967]. See also Don Knuth's work-in-progress section 7.2.2.3 of The Art of Computer Programming (https://www-cs-faculty.stanford.edu/~knuth/fasc7a.ps.gz) for a computational discussion graceful labelings as constraint satisfaction problems.

Proof. We have $\sum_{uv\in E(G)}|l(u)-l(v)|=1+2+\dots+m=\binom{m+1}{2}$ when there are $m$ edges; and a given vertex $v$ appears exactly $\deg(v)$ times in this sum. Working modulo $2$, we also have $|l(u)-l(v)|\equiv l(u)+l(v)$. Therefore $\sum_v\deg(v)l(v)\equiv\binom{m+1}{2}$. But $\binom{m+1}{2}\equiv1$ when $m=4k+1$ or $m=4k+2$. $\square$

Overview: gracefulness of windmill graphs $K_n^{(j)}$

When $n=2$, $K_2^{(j)}=K_{1,j}$ is a star and can always be gracefully labeled by placing $0$ in the internal vertex.

When $n=3$, $K_3^{(j)}$ is a friendship graph and is graceful if and only if $j\equiv0\hbox{ or }1\pmod4$. The forward implication follows from J. Bosák's result in [Rosa 1965], and the reverse implication follows from a construction of [Skolem 1957].

When $n=4$ and $4\le j\le1000$, $K_n^{(j)}$ is graceful due to a construction of [Ge et al. 2010] (they construct $(12j+1,4,1)$-perfect distance families for $4\le j\le1000$, which are equivalent to graceful labelings of $K_4^{(j)}$). When $j>1000$, nothing is known, though [Bermond 1979] conjectures that $K_4^{(j)}$ is graceful for $j\ge4$.

When $n=5$, if $j$ is odd, Bosák's result implies that $K_5^{(j)}$ is ungraceful. For even $j\ge6$, nothing is known.

When $n\ge6$, a result of [Koh et al. 1980] implies that $K_n^{(j)}$ is always ungraceful.

References

[Skolem 1957] Thoralf A. Skolem, On Certain Distributions of Integers In Pairs With Given Differences. Mathematica Scandinavica 5 (1957), 57–68. https://doi.org/10.7146/math.scand.a-10490

[Rosa 1965] Alexander Rosa, O Cyklických Rozkladoch Kompletného Grafu, Kandidátska dizertačná práca. (Bratislava: Českoslovanská akadémia vied, November 1965), ii+86 pages. (Note: Rosa attributes the result above to J. Bosák on page 17). https://archive.org/details/o-cyklickych-rozkladoch-kompletneho-garfu

[Rosa 1967] Alexander Rosa, On certain valuations of the vertices of a graph. Theory of Graphs (International Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967), 349–355. https://www.researchgate.net/publication/244474213_On_certain_valuations_of_the_vertices_of_a_graph

[Bermond 1979] Jean-Claude Bermond, Graceful graphs, radio antennae and French windmills. In Graph Theory and Combinatorics (ed. R. J. Wilson), Research Notes in Mathematics 34 (1979), 18–37. (Proceedings of a one-day conference in combinatorics and graph theory held at the Open University, England, on 12 May 1978.) https://hal.inria.fr/hal-02340680

[Koh et al. 1980] Khee Meng Koh, D. G. Rogers, H. K. Teo, and K. Y. Yap, Graceful graphs: some further results and problems. Congressus Numerantium 29: Proceedings of the 11th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Winnipeg, Manitoba (December 1980), 559–571

[Ge et al. 2010] Gennian Ge, Ying Miao, and Xianwei Sun, Perfect difference families, perfect difference matrices and related combinatorial structures. Journal of Combinatorial Designs 18(6) (2010), 415–449. https://doi.org/10.1002/jcd.20259

[Knuth 2021] Donald E. Knuth, The Art of Computer Programming Volume 4 Pre-Fascicle 7A, Section 7.2.2.3: Constraint Satisfaction (2020–). https://www-cs-faculty.stanford.edu/~knuth/fasc7a.ps.gz

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.

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 $k$. (Proof added below.)

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 section 7.2.2.3 of The Art of Computer Programming (https://www-cs-faculty.stanford.edu/~knuth/fasc7a.ps.gz) for a discussion of computational aspects of graceful labelings as constraint satisfaction problems.

Lemma 7.2.2.3O from The Art of Computer Programming. In any graceful labeling of a graph with $4k+1$ or $4k+2$ edges, the number of vertices with an odd degree and an odd label is always odd.

Proof. We have $\sum_{uv\in E(G)}|l(u)-l(v)|=1+2+\dots+m=\binom{m+1}{2}$ when there are $m$ edges; and a given vertex $v$ appears exactly $\deg(v)$ times in this sum. Working modulo $2$, we also have $|l(u)-l(v)|\equiv l(u)+l(v)$. Therefore $\sum_v\deg(v)l(v)\equiv\binom{m+1}{2}$. But $\binom{m+1}{2}\equiv1$ when $m=4k+1$ or $m=4k+2$. $\square$