Tom Ruane of SABR & Retrosheet extracted this information from the database for me. The answer is the Angels, just last year, on Sept. 30. They lost to the A's 8-7. In the game, the Angels had an OPS of 1.226 while the A's had .567 for a differential of .659. Tom just posted an analysis that goes into alot more detail and covers other issues like what was the highest OPS for a certain number of runs scored in a game, etc. See Fun with a Team's OPS.
The Angels outhit the A's 13-6. The Angels had 5 HRs and a 2B while the A's had 1 2B & 1 3B. The Angels walked 6 times and the A's had 7. The Angels made 4 errors and the A's made 2. All of the runs the A's allowed were earned while only one allowed by the Angels was earned. There were no hit batters in the game.
4 of the 5 HRs the Angels hit were solo shots and the other was a 2-run HR. The A's 2B came in the 4th inning with 2 outs and bases loaded. One of the runners had reached on an error. All three runners scored and all of those runs were unearned. Something similar happened in the 7th inning when the A's scored 4 runs.
Click here see the boxscore of that game
Here is the rest of the top 10. The first number is the OPS differential. Then the team that lost while having that differential followed by the date then the OPS they had and the OPS they allowed.
.643 OAK A 1998-5-10 .877 .234(lose to White Sox 4-3)
.583 MON N 1999-4-19 1.360 .777(lose to Rockies 11-10)
.573 PIT N 1918-6-18 .730 .157(lose to Phillies 1-0)
.526 BRO N 1930-8-11 1.157 .631(lose to Cardinals 7-6)
.523 PIT N 1917-5- 6 .821 .299(lose to Cubs 3-2)
.522 KC A 1979-4-30 1.175 .653(lose to Rangers 8-7)
.505 MIN A 2005-8-31 .899 .394(lose to Royals 1-0)
.492 DET A 1950-8-11(2) .944 .452(lose to Browns 2-1)
.491 TEX A 1989-8-15 .725 .234(lose to Mariners 2-0)