Chronology alone settles the issue of whether there can be a a destruction of the temple in 586/585 BC and therefore the idea that the 37th year of Nebuchadnezzar could be in 568, BC.
Solomon's Temple was dedicated in 950 BC. This alone eliminates the possibility that the VAT can refer to 568 BC.
You have to remember, there are the remaining years of Solomons Reign which approx 28 more years. Then you have the 390 years of the Kings as spoken of by Ezekiel 4. And about 3 years in between before the House of Judah departed. So in all about 420 years from the Dedication of the Temple to its destruction.