Hi, Dennis. I'm glad you enjoyed my book. I'll try to answer both questions, but in two posts, maybe.
Both Chatton and Ockham agree that we have to posit the existence of things; there are philosophical arguments that force us to do this, as well as sensory experiences and religious sources. If I see a chair I'm bound to admit the chair exists; God announces himself in many ways as well. So we must believe in these things.
The more difficult question for theologians arises when we consider the philosophical background we want to adopt or continue with as we do theology -- in short, how much Aristotle should we accept, and how should this Aristotle be understood? Here is where the first thing I mentioned comes in to play -- rational argumentation. Aristotle seems to make us to admit that form and matter exist, that accidents exist, etc., by force of argument. The issue is simply this: how are his claims to be understood, and how good are the arguments for them? For example, maybe we agree that there are qualitative accidents, such as this whiteness in the paper on my desk, but we do not agree whether there is such a thing as the relation of similarity that this whiteness in this paper has with that whiteness in that paper. Aristotle said things that might allow me to believe in the existence of relations and to develop a theory of relations along the lines he provided. But do you really have to accept the existence of these things or not?
In the course of fighting about these issues the English especially became fond of trying to short circuit certain arguments by type, that is, to knock down whole swathes of argument in favor of the existence of certain entities by claiming that alternative explanations could be given for the phenomena those arguments based themselves on. These are razors, if you like. The gist is that you don't need to posit those entities, actually.
This strategy is open to overuse by those zealous in combating the level of Aristotelianism in theology, to the extent that certain very useful parts of that philosophical framework are rejected, the baby becoming lost with the bathwater. And so a backlash emerges, the anti-razors, which attempt to expose the poverty resulting from crippling certain parts of Aristotle.
Ockham was generally keen to reduce and restrict what he regarded as sloppy, thoughtless adoption of Aristotle in theology, and so aimed to short-circuit many arguments he didn't like (arguments for universals, relations and the like). Chatton, attempting to salvage more room for traditional theology and to make a name for himself, went up against Ockham on this general point, reminding him that sentences about things, if they are to be true at all, require us to posit the things they are about as existing. Moreover, and here is the difference between the two men, Chatton thought that once posited, certain entities (like first substances) require you to posit others (like relations) in order to make the system work as a whole. Ockham recognized this general issue under pressure, but tended to argue that we could get away with even less Aristotle, and that we should, because the arguments for certain parts of his philosophy (like first substances) was better that the arguments for others (like relations). Hence Chatton's anti-razor attempts to save and defend a larger chunk of Aristotle for theology than Ockham's anti-razor does.
More technically and specifically, Chatton holds that changes in truth-value of sentences of physics, theology, and philosophy, say, cannot be explained as Ockham often does using connotation theory by citing the passage of time or a particular arrangement of entities as causes of the truth-value change; sometimes, Chatton said, you must posit the arrangements themselves as entities. If a change, say in time or location, make a sentence that was true now false, then mechanically speaking you must posit those entities needed to explain all aspects of the change, and if an arrangement of entities make a sentence that was true now false, then mechanically speaking you must posit those entities needed to explain all aspects of the arrangement. And this might include positing that the arrangement itself is an entity... For example, must we say that there is a relation of similarity inhering in two similarly white pieces of paper, existing as a real, separable thing in those pieces of paper (Chatton), or can we take their similarity as a brute fact, saying that they are similar simply in virtue of their similar whitenesses (Ockham)? Chatton tended to believe that more of the technical machinery of Aristotelian metaphysics would have to be retained in order to explain especially dynamic situations, such as physical change. Ockham's sin, in his eyes, was in stripping the machine down so far that it would fail to perform not only for theology but even for physics.
In Ockham Explained, for example, I try to argue that Chatton is correct in the case of motion. It's not that Aristotle's theory of local motion must be kept in all respects, but simply that, if you want to give a broadly Aristotelian account of motion you cannot reduce your stock of entities as far as Ockham does, the same way you can probably keep me alive if you take out my appendix and my gall bladder, but not my brain.
I love Ockham's rigor and high standards of evidence, but I think Chatton is right that his philosophy is ontologically crippled. He really does try to get away with a tiny, tiny set of metaphysical categories...