There is an error in your analysis. There is no reason to start 6 hours ago, since the alternative is being able to go back 6 hours in the future. Either way, he has the same amount of time. If he finds out he needs to do something in the past, then he could go back. The only difference between researching first and going back first, is that if he researches first, he keeps the option of using the time turner to do something else. (e.g. Use 6 versions of him to do something in parallel at the same time).

When choosing between closing doors available to him and researching, the rationalist researches first.