So I think first of all that you're right to identify Quine as committed to saying something important about how the practice of science relates to our sense of "what there is" in the world. While from his earliest writing Quine was already talking about semantic holism, that our unit of interpretation in empirical hypothesis testing involves the whole mesh of scientific practice rather than any specific individual molecules of meaning, he wanted to hold concepts of logic and mathematics as very tightly centred within that mesh, and playing very important roles in practical scientific communication.
Specifically, Quine believed that to be existentially committed something in our widely communicated idiom is to correctly use existential quantification, as outlined in first order classical model theories, in statements in a language that supports good scientific practice. In "On what there is" (p32), Quine put it thus:
We can very easily involve ourselves in ontological commitments by saying e.g. that there is something (bound variable) which red houses and sunsets have in common, or that there is something which is a prime number between 1000 and 1010. But this is, essentially, the only way we can involve ourselves in ontological commitments - by our use of bound variables. [...]
The variables of quantification, "something", "nothing", "everything", range over our whole ontology, whatever it may be, and we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true.
This assessment doesn't itself serve to resolve disputes about what there is. In that paper Quine went on to talk about the disagreement between Logicists, Intuitionists and Formalist about mathematical objects, where he pointed out (p36) that:
We look to bound variables in connection with ontology not in order to know what there is but in order to know what a given remark or doctrine, ours or someone else's, says there is; and this much is quite properly a problem involving language. But what there is is another question.
In debating over what there is, there are still reasons for operating on a semantical plane. [...] So long as I adhere to my ontology, as opposed to McX's, I cannot allow my bound variables to refer to entities which belong to McX's ontology and not to mine. I can, however, consistently describe our disagreement by characterizing the statements which McX affirms. Provided merely that my ontology countenances linguistic forms, [...] I can talk about McX's sentences.
For Quine, since existence is very particularly a matter of understanding a logical idiom, the paradigm of ontological dispute between two parties with differing conceptual schemes was not so much "deciding who is right", but rather "how discourse can progress". Each party can both retain their own commitments separately from one another, while also understanding the other as making assertions that point to alternative models.
On one level then, we might see the upshot of Quine's ideas as presenting a program for how to construct error theories about different ways of thinking once we know better what the actual truth of the matter is. We as the winners of the dispute, writing the scientific textbooks after we have got together, tested our hypotheses and remained the last novel-predictive theory standing, get to understand those who disagreed with us as presenting a linguistic model that turned out to in fact be false, but which as a set of sentences at the time seemed entirely consistent with the evidence as it was. Our semantics, meanwhile, is grounded in the ontology of our best-fit scientific practice.
And this way of seeing things works out even if we haven't yet emerged victorious - each school can accommodate an error theory of their rivals within their model of reality, as part of their overall conceptual scheme of meaning, without needing to build in specific ontological extensions for their rivals' interpretations. These competing schools might be incompatible, but that's okay - each can develop their own ontology on the basis of what we all know so far, and as time goes on and more experimental data accrues we may well come to a place of interpreting one as better than another and thus able to account for the plausibility of the other.
However, Quine himself came to posit that the problems of interpreting sentences framed relative to other conceptual schemes run much deeper than this. In his book "Word and Object", and more particularly in his paper "Ontological Relativity", Quine treated the analysis of the semantic practices of other speakers as a live empirical project, and the problem emerged that it was no longer exactly clear when we ought to treat assertions as using the idiom of classical logical quantification and thus suited to our model theories.
This he called the "Inscrutibility of Reference", and it gives rise to a need for further "Analytical hypotheses" on our part about how each speaker (/community of speakers/set of abstract linguistic practices) maps on to our own logical connectives in their use of language. Now you might see the infinite regress that threatens: these hypotheses are, themselves, prone to the same conflict that the potential for a plurality of interpretations might bring about. Which metatheory is right? Well, let's try to build a theory of that... Oh, we have to construct more analytic hypotheses to account for this, so we need a metametatheory, etc.
And the pièce de résistance for all this is that we also can't appeal to our own semantics as grounding a theory of meaning suitable for translation, because we also need to think about it in terms of others who "speak the same language as ourselves" - do I and someone else speaking international English have the same ontological commitments? How would I reasonably theorise someone to have the same ontology as myself? What are my semantic posits, and what theories about myself am I positing in order to give such an account?
So by the time of OR, Quine abandoned the idea of a correct theory of meaning in favour of the idiom of the Frame of Reference, borrowing from relativity in Physics. The idea of whether our word 'rabbit' "really means" rabbits was dismissed as meaningless, due to the necessary circularity that would be needed to resolve the otherwise infinite regress of analytic hypotheses. Instead, the project of the translator must be to work strictly relative to an established background language.
I think "Ontological Relativity" thus contains a direct answer that Quine might give to your question:
What makes sense is to say not what the objects of a theory are, absolutely speaking, but how one theory of objects is interpretable or reinterpretable in another. [...]
It is thus meaningless within the theory to say which of the various possible models of our theory-form is our real or intended model. Yet even here we can make sense still of there being many models. For we might be able to show that for each of the models, however un-specifiable, there is bound to be another which is a permutation or perhaps a diminution of the first.
I'd propose that in a position where we are genuinely unsettled as to which of the various interpretations of our theory is to be treated as cannon, one might act in the spirit of Quine's Ontological Relativity to see this indeterminacy as an important feature in the establishment of an interpreting meta-theory, and that one should work in a background conceptual scheme that is strong enough to account for each of the candidate models. We forgo the prior requirement that there should be a determinate fact of the matter as to what each of the theories mean - rather, I should say what the meanings of our theories are - in favour of a relational model that can interpret the different theories in terms of each other.
The upshot of this is that it facilitates a principle of Charity - one is able to build bridges between the different theories, to best enable mutual co-operation in the resolution of their respective disputes, and to thereby accelerate the likelihood that if there is in fact a difference to be found in the empirical data, it will be reached, effectively communicated and the disagreement thus resolved amicably.