Yes. (with the possible exception of how you define “gratuitous”)
The next threshhold is defining “greater good”.
That’s where the problem often lies for humans. They fail to view the “greater good” from a perspective that *encompasses a state of eternal existence * - they see only temporal/earthly consequences. God knows the greater good for each because He can see all of eternity; we cannot.
I accept this for the sake of this discussion. Though I have to remark that the Christians assert that God
is benevolent, and they have no knowledge of those “eternal good things” either. So at best it is an impasse. Of course I can bring up millions of examples which
look like a corollary of a non-benevolent God’s existence.
Mathematical, logically required,

(I’m not sure mercy, love, forgiveness, joy, etc. can be contained under the categories of mathematical and logical!) I’d also change the word “required” to “given”. It isn’t necessarily required; it may be given to us as an opportunity to grow spiritually and to merit greater eternal reward.
No, it cannot be sustituted. No matter how great a “reward” is given, if the prior pain and suffering is
not a logical necessity, the reward is irrelevant.
Please understand, this whole problem stems from the problem of “evil”. God allows pain, suffering and moral evil - that cannot be disputed. It can only be morally justified if the pain, suffering and moral evil is
logically necessary. An unconnected “greater reward” even “infinitely greater reward” cannot justify retroactively the prior suffering.
But I do go along with your premise that God’s “knowledge and power allow Him to mete out the precise amount of pain and suffering to create some greater good.” And, contrary to you, I believe that is exactly what He does.
It is possible. But your stipulation is
contradictory to the Church’s teaching - namely that God rarely ever interferes with our existence and
never with our free will. What you say here is that God
constantly “tinkers” with our free will.
Imagine a father beating his child for no discernible reason. Suppose that a greater good will come out of it, (I cannot see any good, and neither can you), but maybe God does. After the
necessary amount of pain is administered, God
must stop the beating, lest it becomes excessive. And that is interference with the father’s free will.
And there are millons of such actions every second (both moral and natural instances). God must interfere millions of times every second. That is denied by the Church’s doctrine.
Our difference arises when it comes to acknowledging who is that is capable of defining and recognizing what the “greater good” is. I believe it is God (as do most who believe in God).
I think you consider the “greater good” must conform to the boundaries of your concept of reality - and to the line of your reasoning within those boundaries. Please correct me if I’m wrong.
I was willing to stipulate that our understanding is insufficient. And that leads directly to God’s constant interference with our free will. You cannot have your cake and eat it, too.
I know you are atheist, ateista. But your argument concerns God and His involvement in our human history according to Christian belief (I realize it’s only theoretical in your eyes). For an atheist, the highest logic is human logic; for someone who believes in God, there is a far greater logic, God’s “logic”.
There is only one logic. God’s logic is just as oxymoronic as military “intelligence”. God may have more facts to operate with, and a better horizon to evaluate “stuff”, but the laws of logic are the same.
If something is morally evil for humans, then it is also morally evil for God. “Quod licet Iovi, not licet bovi” is nonsense.