"Joyal type" model structure for (n,1)-categories?

I don't know if this will work with the definition of $(n, 1)$-categories exactly as stated by Lurie, but it will work with a reasonable modification.

Definition 2.3.4.1 from HTT basically says that a quasicategory is an $(n, 1)$-category if it has no non-trivial morphisms above dimension $n$ and it insists that this condition is satisfied on the nose. This is probably too much to ask if we aim for a nice model for the homotopy theory of $(n, 1)$-categories. (In particular, the definition is not invariant under categorical equivalences.) We would rather impose this condition homotopically.

It is perfectly sensible to define an $(n, 1)$-category as a quasicategory whose mapping spaces are $(n-1)$-truncated. This can be concisely stated as follows. An $(n, 1)$-category is a quasicategory $\mathcal{C}$ with the right lifting property with respect to inclusions $\partial\Delta[m] \to \Delta[m]$ for all $m \ge n + 2$. I'm pretty sure that this is equivalent to $\mathcal{C}$ being categorically equivalent to an $(n, 1)$-category in the sense of HTT.

Now we can simply take the left Bousfield localization of the Joyal model structure with respect to the maps $\partial\Delta[m] \to \Delta[m]$ for $m \ge n + 2$. This exists by the general theory of Bousfield localizations (see e.g. Hirschhorn's book) and has $(n, 1)$-categories as fibrant objects, monomorphisms as cofibrations and weak equivalences between fibrant objects are exactly the categorical equivalences.

I'm not aware of any reference that constructs this model category, much less one that develops it any further.

You may be interested in having a look at this: https://arxiv.org/abs/1810.11188.