As far as I'm aware, no such conditions is known - The paper of Anel and Lejay is the closest to an answer available in the litterature.
So, this is not an answer to the question, but more of an expanded comment explaining what is the main subtleties to keep in mind when trying to find such conditions. In short, the point of the story is that in order to understand the kind of conditions you are looking for, you need to use a description of your topos using something other than a site.
You can either take it as an explanation of why it might be a misguided question or as an explanation of what a solution need to look like.
The general idea (Which I've learned fron Anel) is the following: (this applies both 1-categorically and infinity-categorically). For a topos $T$ to be exponentiable, which as mentioned in the OP is equivalent to the topos being continuous, it is enough for the exponential $S[O]^T$ to exists, where $S[O]$ is the object classifier, i.e. such that $Hom(T,S[O]) = T$. This means in particular that we need to have a topos whose points are the objects of $T$, so we need the objects of $T$ to admit a description using geometric logic, that is using only condition involving finite limits and arbitrary colimits.
Of course, In general the sheaf condition involve infinite limits - that's why not all topos are exponentiable! But, the problem is that even in the case of a locally compact Haussdorf space the usual description of sheaves still involves infinite limits and is non-geometric.
The trick to get out of this problem is that in some case, there is an alternative way of describing sheaves - what Anel and Lejay called "Leray sheaves". This refers to Leray's original definition of sheaves, which were defined on manifolds only and instead associating to each open a set of sections associated to each compact a set of section. There is an equivalent definition of a sheaf on a locally compact space in terms of its section on all compact subspace, and this definition only involves finite limtis and arbitrary colimits instead of the infinite limits of the usual definition of a sheaf.
This is what makes the notion of sheaf "geometric" and why there is a topos that classifies sheaves on $T$ (that is $S[O]^T$).
So, if you do have finiteness conditions on your topology, then yes, you can deduce from it that the topos is exponentiable, but the converse is a lot more subtle: As soon as you can find some alternative definition of sheaves, potentially using other sets than the set of sections, where the various condition in the definition only involves finite limits, then the topos is exponentiable. But this alternative definition can have very little to do with the site you started from, so it might be very hard to track it back to a condition on the site.
The paper of Anel and Lejay gives the most* general description of what such an alternative description might look like.
(*: by "most" I mean it is a necessary and sufficient condition, so that any other such description should be deducible form there's)
