It seems to be a regular assertion among biblical commentators that the phrase 'house of Israel', in Ezekiel 4, is a reference to the (northern) kingdom of Israel. For the following reasons, those who have taken that position should reconsider.
The (northern) kingdom of Israel was destroyed and carried away by Assyria over 100 years prior to the events of the first chapter of Ezekiel, and were not regathered to the land.
Prior to chapter 4, the phrase 'house of Israel' appears in the book of Ezekiel 5 times. None of those are references to the (northern) 'kingdom of Israel' since God is directing Ezekiel to speak and interact with the 'house of Israel' in his present time.
The phrase 'house of Israel' is used in chapter 4, 3 times. None of those can be reference to the (northern) 'kingdom of Israel' since he is directed to perform 'a sign to the house of Israel'.
After chapter 4, there are another 75 instances of the phrase 'house of Israel' in the book of Ezekiel. None of these are a reference to the (northern) kingdom of Israel.
The phrase 'kingdom of Israel' does not appear in the book of Ezekiel. Neither does 'kingdom of Judah'.
The phrase 'king of Israel' does not appear in the book of Ezekiel. Neither does 'king of Judah'.
The phrase 'house of Israel', throughout the book of Ezekiel, refers to the whole house of Israel, all of the tribes.
The phrase 'house of Judah' is a more specific reference singling out the tribe of Judah.
If what you're suggesting is correct, then we should be able to substitute those terms respectively and it should make sense. Let's see...
Ezekiel 4:
1 Thou also, son of man, take thee a tile, and lay it before thee, and pourtray upon it the city, even Jerusalem:
2 And lay siege against it, and build a fort against it, and cast a mount against it; set the camp also against it, and set battering rams against it round about.
3 Moreover take thou unto thee an iron pan, and set it for a wall of iron between thee and the city: and set thy face against it, and it shall be besieged, and thou shalt lay siege against it. This shall be a sign to the (whole) house of Israel.
4 Lie thou also upon thy left side, and lay the iniquity of the (kingdom of Israel) house of Israel upon it: according to the number of the days that thou shalt lie upon it thou shalt bear their iniquity.
5 For I have laid upon thee the years of their iniquity, according to the number of the days, three hundred and ninety days: so shalt thou bear the iniquity of the (kingdom of Israel) house of Israel.
Not only would the phrase 'house of Israel' suddenly change meaning from the first six mentions in the book of Ezekiel (the whole house of Israel, all of the tribes), to this new meaning, 'kingdom of Israel' for only two mentions, and then back to the original meaning for the rest of the book.
Neither would it make sense that 390 years of iniquity are attributed to a kingdom that only existed for ~240 years, and had been carried away and dispersed for over 100 years.
On the other hand, if the phrase 'house of Israel' keeps it's original meaning (the whole house of Israel, all of the tribes), then it not only remains consistent throughout the book of Ezekiel, but it retains its logic. According to your own admission, the great Bible scholar, Edwin R. Thiele, was only ~10 years off (380/390) of what God attributed to the 'house of Israel' (the whole house of Israel, all of the tribes) as their years of iniquity.
It also means that the sign that God was telling Ezekiel to convey was actually going to those guilty of the iniquity, but also present and able to receive it... even though He knew that they would reject it.
We should agree that James 1:5 provides us with a greater source of truth than tradition; taking into account also, that there have been many scriptural matters that were sealed until this generation.