Let's say I have a small category $A$. Does there in general exist a right Bousfield localisation of simplicial presheaves on $A$ with the injective model strucure, where the weak equivalences are created by homotopy limits? What if I make extra assumptions, such as $A$ having a contractible classifying space or being a test category?

'sometimes'

The problem with right-bousfield localizations is that presentable ∞-categories aer an inherently handed-notion

lol okay trying to fix some silly paper of mine about ∞-categories and it's been quite a while since I've thought about this type of thing... in Lurie's framework, "lax monoidal" is a map of ∞-operads between monoidal ∞-categories right?

It's a relative operad map over Assoc.

Yeah right okay... by defining algebras as sections that preserve the inert morphisms, an ∞-operad map, which preserves inert morphisms by definition, necessarily takes algebras to algebras...