Practical numbers with tiny representations: is $h(m)<(\log\log m)^{O(1)}$ infinitely often? (Erdős #18)

Call $m$ practical if every integer $1\leq n<m$ is a sum of distinct divisors of $m$. For practical $m$, let $h(m)$ be the least $k$ such that every integer $1\leq n<m$ is a sum of at most $k$ distinct divisors of $m$. Are there infinitely many practical $m$ such that $$h(m)<(\log\log m)^{O(1)}?$$ Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$?

Acceptance. FULLY RESOLVES: a proof that there are infinitely many practical $m$ with $h(m)\leq(\log\log m)^{C}$ for some absolute constant $C$ (this is the $250 question), or a proof that no such infinite family exists; and/or a proof or disproof of $h(n!)<n^{o(1)}$ (or the stronger $h(n!)<(\log n)^{O(1)}$). Machine-checkable (Lean/Coq) preferred, else a complete written proof. ADVANCES: (a) an infinite family of practical $m$ with $h(m)$ of strictly smaller order than the $(\log m)^{1/2}$ bound stated in the background, with proof; (b) an upper bound for $h(n!)$ asymptotically better than the $h(n!)<n$ bound stated in the background, with proof; (c) a nontrivial general lower bound on $h(m)$ for practical $m$, with proof; (d) exact values of $h(m)$ for all practical $m$ up to a substantial bound, or of $h(n!)$ for as many $n$ as feasible, with reproducible search code, per-value certificates (a worst-case target $n$ and an optimal representation), and an analysis of the empirical growth rate; (e) a Lean formalization of Vose's theorem. Deliver the proof file, or the code + certified value tables.