Sort of, but that would be immune to the strongest part of my argument; which involves the children. We could dispute plausbly either way if, for example, there were any healthy adults which could be held to be an Amalekite proper and I am willing to concede, given the seemingly identity relation between being an Amalekate and a part of the cult, that there weren't any. — Bob Ross 
Okay, that's fair.  I just wanted to try to impress the idea that the Amalekite culture and the Amalekite religion/cult go hand in hand, and if we want to get into the exegesis we could show that it is specifically the abominations associated with the Amalekites that God is concerned with.  The question, "Why the Amalekites?," is something we ought to keep in mind.  It would be a significant mistake to assume that this is how God/Israel deals with 
every people-group.  But let's move on to children.
At the end of the day, I emphasize the children, although I understand you are setting that aspect of it aside for a second, because it is really implausible in my mind that there were no Amalekate children and it seems like they would be a part of the ban. — Bob Ross 
I think it is reasonable to assume that there were Amalekite children and that they were part of the ban.
1. The God of the OT commanded Saul to put the Amalekites under the ban
2. There were innocent children among the Amalekites
3. Therefore, the God of the OT commanded the killing of the innocent
4. The killing of the innocent is unjust
5. Therefore, the God of the OT is unjust — Leontiskos 
In the first place I would want to note that in our Western society which strongly values individualism, the individual is the central agent and the child is often seen to be his own person, so to speak.  I saw the new 
Superman movie (which I did not think was very good) and there is a scene where Clark's father is telling him that parents don't shape their children's lives, but instead give the children tools with which to shape their own lives.  That a pretty standard individualistic sentiment, and it would in no way have been the view of ancient peoples.
To oversimplify, the ancient world is going to see the child as strongly shaped by their environment—both "nature" and "nurture"—whereas our own culture tends to see the child as a free agent who largely transcends their environment.  I think we have veered too far in the "libertarian" direction, and I think that a factual or statistical analysis would show that children are deeply influenced by environment and culture.
A second consideration is the question of support mechanism.  Suppose Israel wipes out the adults.  Would they have the resources to absorb all of the children into their own numbers?  That seems unlikely, and neither is it clear that the children would be overly cooperative at that point or even when they grow older.  So there is the simple logistical problem, where there is a people-group who practices abominations (human sacrifice, cannibalism, rape, demon worship, etc.) and you have to address the problem.  How do you address it?  Given that the adults are not able to be reformed, they must be imprisoned or killed, and imprisonment of such a large number would have been impractical in that day (if not in ours as well!).  So what do you do with the children?  How do you view the children?  Similarly, what is 
best for the children?  Should they be left to live without parents and support?  Should they be left to grow up into evil cannibals (in the case where their parents are not killed)?  Should they be abandoned to their fate if they cannot be incorporated and supported?  I don't see any obvious answers here.  Indeed, the command to kill the children is much like a command to pull out the weed by its root, so that it does not regrow.
Now your argument is apparently thinking in terms of commutative justice, where the child is the agent, the agent has done nothing wrong and is therefore innocent, and therefore the child cannot be harmed and certainly not killed.
So at this stage we have three considerations which cannot be altogether ignored:
- Individual agency vs. group agency
- How to address the problem of abominations which have become embedded in a people-group
- The injustice of killing the individual, including children
The injustice argument has a certain preeminence given that it is trading in exceptionless norms.  More explicitly, 
if the Amalekite children have a right to life, then it is unjust to kill them.  So we probably want to ask whether they do in fact have a right to life, even though they are 
Amalekite children.
Certainly if we think of agency in terms of groups instead of in terms of individuals, then it is no longer clear that the Amalekite children have a right to life.  More specifically, it is no longer clear that the Amalekite children are innocent, given that they are inextricably bound up with an abominable group.
Note that when thinking in terms of group agency rather than individual agency, children of the Edomites, for example, 
are innocent in virtue of their people-group and therefore do have a right to life.  Or more simply, the commandment against murder applies straightforwardly to them.  So the criterion of innocence has not been abandoned, but is rather being interpreted and applied differently.
Anyway, those are three of the basic data points I think we would need to consider when thinking about the Amalekite children.
If so, then how do you explain the fact that God punished Saul for sparing some animals? Doesn't that suggest that God was including everything that lived in the City itself? — Bob Ross 
At the very end of that clip I suggested this is addressed quite well (beginning at 
1:11:45).  If you didn't get a chance to watch 
those 18 minutes I would recommend it.
P.S. The reason you aren't getting a lot of direct answers to your argument in this thread is simply because it is a very difficult argument to address.  For that reason I'm not sure whether I will succeed in giving you a satisfactory answer either, but I think these considerations complicate the initial picture quite a bit.