The view you are suggesting is something close to what is held by
Solomon Feferman, who holds that the objects and truths of
arithmetic have a definite nature that is not shared when one
moves up to higher-order objects, such as the collection of all
sets of natural numbers. Feferman has long been known for the view that the continuum hypothesis is inherently vague, in a way that arithmetic is not, and this seems to be basically what you are talking about. See for example his article
There are several other papers for the EFI project exploring similar issues.
One interesting aspect of the view is the idea of using classical logic in the lower more-definite realm, and intuitionistic logic in the higher realm, where assertions such as the continuum hypothesis may have a less definite meaning. Nik Weaver has pointed out in the comments below that he had first proposed this dichotomizing idea in his 2005 article:
Finally, let me criticize your use of the term
Platonism to imply a kind of singularist view of mathematical existence, whereas I have argued that it should instead imply only a kind of realism or definite existence. With this idea, the multiverse view itself is a kind of Platonism, where one gives up on the uniqueness of the existence of
mathematical objects, but not on their objective existence. For example, on the
multiverse view in set theory, there are many different concepts
of set, each giving rise to its own set-theoretic universe, which are just as real as the set theory claimed by the universists.