The account of David's son from 2 Samuel is apt indeed, but not for the reasons you seem to suggest.
Your summary statement quoted above is certainly true, but it ignores the central question of petitionary prayer for others if one accepts the determinist model. The determinist must assume that David was a fool to pray to God in the first instance since the death (or life) of his son was (in the determinist presupposition) entirely foreordained and, as you underline in your quote, in this unusual instance, Scripture tells us that "the Lord struck the child ..." [It is unclear from the account whether David knew that, in this instance, it was the Lord who had struck the child. However, in view of David's prior conduct, I think he must have had at least a suspicion.]
David was clearly no determinist, for a good believer in the determinist model would 'know' that God was incapable of changing His mind and thus to pray as diligently as David did would be foolish indeed. But David knew more Scripture and knew that it was possible that God would relent, as He did with Hezekiah and with Niniveh. The fact that He didn't change His mind here doesn't change the fact that David's actions show that he knew that God could and sometimes did do it.
So, for purposes of our discussion here of the determinist model's adverse effect on petitionary prayer, the important point is that David knew that it was possible that God could change His mind.
As to your comments on your father's illness and the conflict in family prayers, I am sure you realize that this is not unusual. The example usually given is that of WWI when there were Christian nations on both sides or our own Civil War with a similar situation. God, however, is all wise and as you suggest can be trusted to sort those things out.
The critical point not to miss, however, is that God is active in human affairs and is responding to those prayers. While as Garth Brooks said in his famous song "Some of God's greatest gifts are unanswered prayers," to take that and convert it, as the determinist must, to a universal that God never answers prayer or never changes His mind is illogical and, worse, unscriptural. Don't you agree?
The determinist must assume that David was a fool to pray to God in the first instance since the death (or life) of his son was (in the determinist presupposition) entirely foreordained
You mean the same foolish David that wrote Psalm 139, verse 16 of which reads, "Your eyes saw my substance, being yet unformed. And in Your book they all were written, The days fashioned for me, When as yet there were none of them."? David most assuredly understood that God also numbered Hezekiah's days.
From God's perspective, no change of mind took place that altered what was already ordained for Hezekiah. However, Hezekiah was changed by praying.
the determinist model's adverse effect on petitionary prayer
If I view prayer as changing/preparing me, how would believing that God is in control of everything, have an adverse effect? It should, rather, spur me. "God does not ask us to tell Him our needs that He may learn about them, but in order that we may be capable of receiving what He is preparing to give."- Augustine
The critical point not to miss, however, is that God is active in human affairs and is responding to those prayers.
Yes, I absolutely agree, because those prayers were foreordained. As David says in verse 4, also from Psalm 139, "For there is not a word on my tongue, But behold, O Lord, You know it altogether.
Garth Brooks said in his famous song "Some of God's greatest gifts are unanswered prayers,"
Garth is a successful entertainer, but not even close as a theologian
God always answers the prayers of His children because He hears them. The answer is not always what they want, but it is always what He wants, because He has ordained whatever shall come to pass. God does not hear the prayers of the unrepentant and so they are unanswered.
"What about Jonah and Ninevah? The argument suggests that God threatened Ninevah with destruction, the Ninevites talked God out of it by prayer, and so God did not carry out His threat. This misunderstands what really happened. For one thing, if the Lord had unconditionally promised to destroy Ninevah, it would certainly have been destroyed. The fact that God did not carry out the promise tells us that it was a conditional promise. Scripture gives us several such examples. It was as if God had said through Jonah, "I will destroy you unless you repent" (Luke 13:3). As the book unfolds, we see that the Ninevites did in fact repent, so they were spared. It was certainly not because Jonah prayed for them. Had they not repented, God would certainly have destroyed them like Sodom and Gomorrah. Notice that they prayed with a view to the sovereignty of God in 3:9. They threw themselves on God's mercy, which is the appropriate attitude of prayer. Thus, they showed that they submitted to his authority, His sovereignty. His revealed will. Look deeply and you will see that God gave them that repentance and faith in the first place. That was because He had ordained to do so. It was not in answer to Jonah's prayer, because Jonah did not want them to be spared." From Curt Daniel, Prayer and the Sovereignty of God
Even their prayer of repentance was a gift from God, He did not change His mind, but the Ninevites were changed.
BTW, I have been praying for you and your friends. It was ordained for me to do that. 8)