Here's the report in War of the Rebellion by General Doubleday, which says Barksdale died on the third day. On a quick search I found other references in War of the Rebellion to his death, but none to the date, but I suspect I missed some. This report says he was wounded in the evening of the 2nd, but of course that doesn't mean he died on the 2nd.
I found reports of his death on the 3rd online, but no sources (see for example 'Gettysburg off the beaten path').
Hmm. Heitman's register (available on Fold3) says that he was 'killed 2 July 1863 at Gettysburg Pa.'
The Wikipedia page says that he died the next morning, but doesn't cite a source.
I checked the 148th Pennsylvania's regimental history, which does have a story about Barksdale's death, but doesn't say whether he died before or after midnight.
Sorry--I thought it would be easier than this to find the answer!