Harnshorne based his process theism on a plexus of arguments, 'a global argument', of which the ontological was only one element and which he treated in a non-standard way. The following extract from David F. Haight might help make things clearer :
The book Charles Hartshorne and the Existence of God, by Donald Wayne Viney, is an admirably lucid, readable and accurate explication of Hartshorne's philosophy of God. Its originality consists in an attempt to explain and defend Hartshorne's "global argument" for divine being - a combination of six arguments, which together are supposed to mutually support and make coherent a process view of the Divine. The global argument consists of a version of each of the following theological arguments: the ontological, the cosmological, the design, the epistemic, the moral, and the aesthetic argument, each of which is the subject of a chapter under that title. Viney elucidates these arguments all very well. I have no quarrel with the fidelity of his interpretations (nor, more importantly, has Hartshorne who wrote the preface congratulating the author). ...
If Saint Anselm's "discovery," according to Hartshorne, is that God necessarily exists - if the concept of God is coherently conceivable (Hartshorne admits that it may not be) - Hartshorne's "discovery" (like A.N. Whitehead's) is that God contains potentiality for growth just like everything else in the universe. (This, for Hartshorne, removes some of the possible incoherence from the concept.) God is no exception to the metaphysical principle of Creativity according to which all events are both active and passive at the same time. As opposed to classical theists (e.g., Anselm, Aquinas, Descartes, Kant, etc.), who hold that God is Pure Act with no potency for change or creative interaction with anything else, Hartshorne claims that God is di-polar and changeable, namely, infinite and finite, necessary and contingent, cause and effect, etc., each contrary being true with respect to a different aspect, or "pole," of Divinity. Hartshorne calls this the "divine relativity." Even though God is a necessary being and cannot fail to exist, the way that S/He is actualized is contingent upon, or relative to, the ongoing process of creation. One benefit that Hartshorne claims for this revolutionary view of God is that evil and suffering are no longer serious problems since God is not all- powerful even though S/He is all-good. Another is that time and death also do not create embarrassments the way they do when God is conceived of as eternal, im- mutable, simple, and all- everything, as in the classical view. After all, if God is all- pervasive, omni-present, timeless, omnipotent and all-good why is there death, suffering and time at all? Only a God who is both infinite in certain respects and finite in others can "save the appearances" of the real world of everlastingness and change, suffering and redemption, life and death, freedom and causality, etc. Moreover, Hartshorne argues, contrary to classicism, that there can be no creation out of nothing, thereby making the cosmos everlasting, which he claims is an a priori truth. Finally, as if much of this were not revolutionary enough, Hartshorne, unlike most theological thinkers, argues that all six of his arguments for God are not em- pirical but a priori, that is to say, are prior to all sensory experience whatsoever and are therefore necessarily true of all such experiences. Such in brief, is the sub- stance of Hartshorne 's neoclassical revisionism, as made clear in Viney's book. (David F. Haight, ' Charles Hartshorne and the Existence of God by Donald Wayne Viney', International Journal for Philosophy of Religion, Vol. 20, No. 1 (1986), pp. 49-53: 49-50.)
From this account I think you will be able to see that Hartshorne did not rely solely on the ontological argument and that God is 'di-polar' and changeable in a way the traditional ontological argument does not recognise.
In this book I do not conceal my own faith -- shared with those just mentioned [he mentioned Plato, Schelling, Fechner, Whitehead, Montague, Berdyaev, Niebuhr] -- that theistic religion thus reformulated [as surrelitivism] is true as well as conceivable; but the only bearing of this personal religious belief upon the argument is that it may afford some evidence that theism can avoid logical absurdities and still be a religious doctrine. [My commends are in brackets.]
Based on the above one may assume what he argues regarding "surrelativism" in this book (and perhaps in his other books) is his "personal justification for theism" presented to avoid logical absurdities.
However, that his modal formulation of Anselm's ontological argument was not what justified his faith makes one wonder why wouldn't it? Donald Wayne Viney writes in Charles Hartshorne and the Existence of God (page 140):
Anselm's ontological argument stands as a reminder of the inadequacy of any merely empirical approach to God's existence. One must distinguish, however, the bare fact (if it is a fact) that God exists and the particular manner in which we conceive of that existence. A particular concept of God may be shown to be inconsistent with empirically discoverable facts. This is not a proof that God does not exist. It is only proof that one conception of the divine existence is inadequate.
This suggests that Hartshorne's faith is based on his concept of God rather than proof of the existence of God.
Viney, D. W. (1997). Charles Hartshorne and the existence of God. SUNY Press.
Hartshorne, C. (1948). The divine relativity: A social conception of God (Vol. 109). Yale University Press. https://archive.org/details/in.ernet.dli.2015.90364/page/n5