The American, son of 1982 U.S. Masters champion Craig, shot a three-under-par 68 to record a 10-under total of 132, three clear of Frenchman Victor Riu (67).
After not dropping a stroke on his way to an opening 64, Stadler was on track for a repeat before he bogeyed his 32nd hole, the par-four fifth.
He birdied the next two before a poor approach on his final hole meant he ended with a double-bogey seven.
"It was not exactly how I planned to finish but overall it was another good day," the burly 35-year-old told reporters."I putted brilliantly the first couple of days. I got a little sloppy on my final nine today but other than that struck the ball very well, had a lot of chances and rolled in a lot of nice putts."
Riu missed the cut in five of his last six events but the Parisian took advantage of some local knowledge to move into contention at the venue of the 2018 Ryder Cup.
Germany's Marcel Siem (72) was third on 137, two ahead of a group of seven players that included defending champion Graeme McDowell (69).
(Reporting by Josh Reich, editing by Tony Jimenez)
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