I agree, he had more than enough time to dispose of the bike and of Mickey. Why the time change would make it any more likely that he brought her to New Orleans escapes me. I think he left Mickey far behind within 1-2 hours of abducting her, personally. JMO
I really can't imagine any reason why Lavergne would choose to bring Shunick (in any condition) to an urban area where she'd be more likely to be found.
Given that Lavergne is supposed to have burned his truck in Texas, dumped Shunick's bike about an hour outside of Lafayette just off I-10, and traveled at least 2.5 hours from his home to visit the hospital, I get the impression that he thought distance might make his actions more difficult to pinpoint.
Everything I've read points to Lavergne being present at Whiskey Bay
around the time the bike was discovered, not around the time Shunick vanished.
Of the 13.5 hours between the time Shunick vanished and Lavergne showed up at Ochsner, only about four of those hours were dark. The sun rose in Lafayette at 6:12AM on May 19th, so logically, he either dumped Shunick between 2-6AM, or stored her somewhere until a later time. I'm not sure how comfortable Lavergne would have been leaving a live, fighting woman unattended for a 10 plus-hour stretch.
Of course it's possible that he dumped the body during daylight hours, but that seems pretty unlikely. On the other hand, it does seem likely that Lavergne's injuries were inflicted in an altercation with Shunick. If he's telling the medical staff and police that his injuries were inflicted around 3PM, then shows up at the hospital about a half hour later, you'd think that the wounds would have to be pretty fresh, and he'd still be wearing the same blood-soaked clothing. The blood wouldn't have had enough time to dry if Lavergne's account of the timing is accurate.
I'm rambling, but I guess the point I'm trying to make is that the window of Shunick's likely death could really be narrowed down by the state of Lavergne's injuries at 3:30PM on May 19th. No doubt the interested parties have obtained the records and are establishing that now, if they haven't already. The more fresh the wounds, the more likely Shunick will be found between Lafayette and Jefferson, LA.