Here is a suggestion that may not be liked much, but has worked well for strict versions of $\infty$-groupoids and categories, namely to use the cubical model.

In

R. Brown, F.A. Al-Agl, R. Steiner, `Multiple categories: the
equivalence between a globular and cubical approach', *Advances in
Mathematics*, 170 (2002) 71-118.

we showed that strict globular $\omega$-categories are equivalent to strict cubical $\omega$-categories with connections. Now (see section 10) the monoidal closed structure on the latter category is not so hard to write down, following the groupoid version given in 1987 by myself and Higgins.

R. Brown and P.J. Higgins, ``Tensor products and homotopies for
$\omega$-groupoids and crossed complexes'', *J. Pure Appl.
Alg.* 47 (1987) 1-33.

Basically this uses the simple idea that for the $n$-cube $I^n$ we have the formula $I^m \times I^n \cong I^{m+n}$.

So the above equivalence of categories allows one to translate the closed monoidal structure in the cubical case to one on strict globular $\omega$-categories. This is related to earlier versions by Street and by Crans.

So the question is: can one use similar cubical methods for weak $\infty$-structures?

The other main advantage of cubical methods for the work with Philip Higgins was the easy description of multiple compositions as arrays of the form $[\alpha_{(r)}]$ where $(r)$ is a multi-index, so allowing *algebraic inverses to subdivision*.