Farrer's view on arguments for God's existence certainly changed during his lifetime and in his published writings. In his early Finite and Infinite (1943) he plumps for a version of Aquinas's argument from the distinction between essence and existence.
(1) Everything around us is a certain kind of thing (has an essence).
(2) Some things - above all, people - are organised so as to have a kind of autonomous activity. Our existence isn't just a collection of qualities, but is something active, free and responsible.
(3) The kind of thing that we are (essence) is obviously distinct from the fact that we are. We don't find anything necessary in our being active beings because we aren't the sole cause of our own existence. (Our essence is distinct from our existence.)
(4) We see it to be necessary that God's essence is His existence, when we understand that God is the ground of our being and of everything else's being. He is the one in Whom existing and the kind of thing that He is are not separate. What He is, is necessary existence. It is in we His creatures that existence and essence are separable.
What Farrer presents in Finite and Infinite is more or less a version of Aquinas's cosmological argument. Later on, he seems to abandon this kind of argument, and appears to be content to ground the idea of God in the faith and experience of the believer - particularly in Faith and Speculation (1967). Thus the claim that he moved from a religious rationalism to moderate fideism.
I wonder, however, if Farrer's later writings point to the potential for the construction of a different kind of argument, which I shall call for convenience the "personal argument" for God's existence.What Farrer works out in Finite and Infinite is a very detailed account of personal existence: he then fails to carry this over into his conclusions when he goes back to apply this account to arguments for God's existence. But what if he had? Then, I think the argument would come out something like this.
(1) We don't experience ourselves as the cause of our own existence: we are called into personal existence by confrontation with and communication with another person who is (so to speak) already there.
(2) This person is in turn, not the ground of their own personhood, so we look to them in vain for the ground of our personhood, even if they have awakened us to a consciousness of our own personal existence.
(3) What we require for an adequate account of personal existence is Someone who is the ground and unity both of the I that is awakened, and the Thou which awakens us. There is a personal meaning which is there prior to either of us, and one knows this experientially but beyond our consciousness as "the Centre, the Fund, or Bottom of the soul" (William Law).
I would like to say that here is an inchoate germ of a "personal" argument for God's existence. But why would it be a preferable argument to any other? I think there are three main reasons. The first is that, unlike the cosmological argument, it requires a personal acknowledgment of a God as the unity and ground of one's own existence, rather than the merely intellectual acceptance of an impersonal Cause. One of the standard Reformed criticisms of the cosmological argument is its impersonal form: it does not lead, it is argued, to the God of Abraham, Isaac and Jacob. Second, and because of this demand for personal acknowledgment in the argument, it is in some sense an evangelical proof, rather than a discursively rational one. The rationality of the proof is embedded in the personal demand for recognition of One upon whom one's very selfhood depends. Part of its certainty (as opposed to probability, the bugbear of Vatican I) is dependent on "giving in" to God's ownership of one's existence. And thirdly, it is a proof grounded in "natural" reason, and therefore satisfies the intuition that our rationality should point back to God as its source if it is in fact His creature. I would argue that one cannot hope to find a more universal "natural" rationality than personal existence.
I wonder - allowing that the argument as I have sketched it needs a lot of work and fleshing out - if this counts as a piece of intellectual Anglican patrimony? It seems to me that the late Farrer, in particular, was a thoroughly, typically Anglican thinker in the best sense. His later notion of rationality was less a discursive, scholastic view of reason, and much more an idea of reason embedded in human culture, and in faith and tradition. It is that particular heritage that I think worthwhile to take from Farrer.