I guess I will just have to stick with the opinion of the nine pastors and commentators who hold to the traditional view. According to the great Bible scholar, Edwin R. Thiele, there were 344 years from the start of Rehoboam's reign to the destruction of Jerusalem and the Temple. Solomon ruled for 40 years, so his fourth year would make 36 more years, which adds up to 380 years, not 390 years.
God had warned Judah about the coming judgment long before Josiah's reign. He even warned Josiah's great grandfather, Hezekiah. "Behold, the days come, that all that is in thine house, and that which thy fathers have laid up in store unto this day, shall be carried into Babylon: nothing shall be left, saith the Lord. And of thy sons that shall issue from thee, which thou shalt beget, shall they take away; and they shall be eunuchs in the palace of the king of Babylon." (2 Kings 20:17-18).
The "whole house of Israel" (all 12 tribes) didn't begin anew during Josiah's reign but during Hezekiah's reign when he invited the survivors of the northern kingdom of Israel (10 tribes) to come to Jerusalem and attend the Passover Feast in the second month (as there wasn't enough time to celebrate it in the first month).
"So they established a decree to make proclamation throughout all Israel, from Beersheba even to Dan, that they should come to keep the passover unto the Lord God of Israel at Jerusalem: for they had not done it of a long time in such sort as it was written. So the posts went with the letters from the king and his princes throughout all Israel and Judah, and according to the commandment of the king, saying, Ye children of Israel, turn again unto the Lord God of Abraham, Isaac, and Israel, and he will return to the remnant of you, that are escaped out of the hand of the kings of Assyria.
For if ye turn again unto the Lord, your brethren and your children shall find compassion before them that lead them captive, so that they shall come again into this land: for the Lord your God is gracious and merciful, and will not turn away his face from you, if ye return unto him.
So the posts passed from city to city through the country of Ephraim and Manasseh even unto Zebulun: but they laughed them to scorn, and mocked them.
Nevertheless divers of Asher and Manasseh and of Zebulun humbled themselves, and came to Jerusalem." 2 Chronicles 30:5-6,9-11).
Yes, God is referring to Judah as the house of Israel in Ezekiel 4:3 as it is all that remains of the two kingdoms, but then He separates the days/years of iniquity between the two kingdoms, Israel and Judah, in verses 5 and 6. The difference in the lengths of iniquity, 390 years compared to 40 years, is evident. Israel's sin was 10 times as great as Judah's was. I have a breakdown of the 390 years for Israel and the 40 years for Judah in my article, The 390 and 40 Years of Ezekiel 4:4-6 – Part II (rev310.net).
Likewise, in Ezekiel 37:16-19, God is referring to the two kingdoms, or houses, of Judah (two tribes) and Israel (10 tribes) coming together in the future and becoming one kingdom again as it was in the days of David and Solomon.