Let $G$ be a multigraph with maximum edge multiplicity $t$ and minimum edge multiplicity $1$ (so that there is at least one 'ordinary' edge).

Is there some simple graph $H$ such that the $t$-fold multigraph $H^{(t)}$ edge-decomposes into copies of $G$?

I believe that Wilson's theory (under certain hypotheses) gives a solution when $H$ is a certain very large complete graph. For instance, see Draganova, Mutoh, Wilson (2008). Heavy machinery gets used there.

Is there a more direct argument if I don't care what $H$ is or how large it is?