Your long question suggested at first that you were asking for something difficult (a locally full internal topos), but you ended up asking for the easy thing.
Starting from your postscript, the purpose of an arithmetic universe is that it is is the least categorical structure in which one can construct the free or initial structure of a widely applicable kind, namely essentially algebraic.
The structure of an elementary topos (with natural numbers if you wish) is essentially algebraic, so there is an initial elementary topos with natural numbers in any arithmetic universe, in particular in any elementary topos with natural numbers.
I would like to refer you to the books 2-Categories for the Working Categorist and Introduction to Arithmetic Universes, but regrettably nobody has written them yet.
Talking about ZFC or cumulative hierarchies in this is really obfuscation. However, I indicated how to build the set-theoretic hierarchy in my paper Intuitionistic Sets and Ordinals, based on earlier ideas of Gerhard Osius.