It's true that there is no way to disprove the existence of higher time chains, implementing our chain the way that we implement Bitcoin. But there also isn't a reason to think there is, from a principle of least action perspective. (And PLA is maybe the strongest ordering principle we can see.)

I think we can get a tiny bit further on first principles reason. I'm cheating though, because St. Thomas already did the work. :)

Namely, we can say, whether there are higher chains or not, that outside the final chain is eternity.
Eternity > Time

Time requires eternity and higher order must exist for the order within the chain for the chain to be intelligible. The effect cannot be greater than the cause.

We exist in eternity, by definition, and we grasp it directly, or we could not make sense of it's approximations here.

Think about this: can you implement anything infinite or even truly continuous inside finite/discrete physical media? But you can implement finite time within eternity.

The trouble with self-cause is that it's turtles all the way down. Only God can be the uncaused cause, by definition. He can't be fully comprehended by us from within the chain, as you say. (Though He could obviously take direct control of utxos of energy inside the chain. Christians believe that He did. The Incarnation.)

Determinism is tricky because from outside the chain, God can see the whole chain and He is it's ultimate and necessary cause. Our perspective in discerning determinism from withing the chain is foreknowledge. If we can work it forward and backward in time, causally, we know that something is deterministic. So when we apply that rubric to imagining the perspective from above for God or eternal created beings in the Aevum, it looks deterministic to us, looking up, as it were.

But the degree of freedom within the chain is whatever is allowed from above. It is not pre determined from withing causal time (from a within the horizontal, directional order). Horizontal determinism applied to vertical determinism is, well, orthogonal.

The Thomistic take is that vertical beings in the aevum are also free, but being outside time their will was instantaneous (for lack of a better word) and eternal. Fixed from our perspecitve. The ordering principles either accepted or rejected the will of God upon their creation. These are what we call angels and fallen angels or daemons.

The rational human intellect and it's grasp of good and evil are proof from first principles of free will. Arguing that the privation of the good (disorder) exists with the allowance of free will is to say that God is not things that He, definitionally and necessarily is: order itself, perfect, unitary, not contradictory, truth itself.

The simpler way I said this to my little boy is "God allows us to do evil because if we couldn't choose and we couldn't be good, and God wouldn't be good if He did that, and that wouldn't be God."

So I think we can say these things from first principles: God exists. He is the uncaused cause. He is intelligent. He creates the chain and is the source of its order from above (causally). The world depends on Him and not the other way around. He created us with rational intellects and free will.

I am probably doing a horrible job here. I apologize for my insufficiency. If this was easy the Summa would be shorter and easy to read. :)

...I wonder if @Neal could possibly be enticed to correct me? (Be gentle.)