Love that story. Using retrosheet, I tried to find the game in question.
Brown did hit a double during a game in early August (8/6, 2nd game of a doubleheader)... but he was in the starting lineup in that game.
Looking further, I assume the following elements of the story are true:
-- Brown hit a double
-- As a pinch hitter
-- In an earlier inning than usual, or in a situation (perhaps with Detroit way ahead?) in which he would not expect to be used.
Based on the story as described, it would ALSO seem to be a game in which Norm Cash was either not in the starting lineup, or had been removed from the game already.
Brown hit 7 doubles that season, so it isn't hard to check all the possible games. In two of them he was in the starting lineup (9/23, in addition to the 8/6 game referenced above). So those would seem to be ruled out.
The other 5 games:
6/3 at Boston: Brown pinch hit leading off the 9th with Detroit trailing 4-3. That doesn't seem to fit.
6/9 vs Cleveland: Brown pinch hit in the bottom of the 8th with Detroit trailing 2-0. That doesn't seem to fit.
7/21 vs Baltimore, 1st game of DH: Brown pinch hit in the bottom of the 7th with Detroit trailing 4-2. That doesn't seem to fit.
8/25 at New York, 1st game of DH: Brown pinch hit in the top of the 8th with Detroit trailing 6-5. That doesn't seem to fit.
9/20 at Washington: Brown pinch hit in the top of the 8th with 2 outs after Detroit had scored 4 runs to rally from a 3-2 deficit to take a 6-3 lead. That COULD be the game, but Norm Cash had just scored in the inning and (after one more batter made an out) went out to play the field in the 9th, so that doesn't quite fit either. But it is probably the most logical game for such a scenario. The Tigers had clinched the pennant, and had just gone ahead. Perhaps he figured Smith would leave Lolich in to finish the non-essential game, so maybe he ordered a celebratory hot dog after his team had taken the lead, thinking he might not be needed.