The Ten Commandments include "Thou shalt have no other gods before me" as if the other gods exist, but Yahweh must be honored and worshipped above the others.
Carthage was a colony of the Phoenician city Tyre--a lot of Carthaginian personal names have the element -bal in them (Hannibal, Hasdrubal, etc.), which is the same as the Hebrew Ba'al.
The only part where I think I disagree with you is the suggestion that the name Baal was not associated with a false god until after Saul's time. The Israelites strayed after Baal even before entering the promised land (Numbers 25); continued after they were in Canaan, before Saul's time (Judges 3:7, many others); and were still committing the same sin after Saul's time (I Kings 18:16-40). So if you're implying that Saul didn't know what he was doing when he gave one son a YHWH-based name and one son a Baal-based name, I have to disagree.
Sorry, FRiend, but politicians weren't any different then than now. Saul was the first King the Israelities ever had, and we know from elsewhere in I Samuel that he was concerned (paranoid, in fact) about being replaced or overthrown. So I believe he was trying to curry favor with every element of the population. Hence the redactors' "shame."