No, the Twins should not trade
While it's likely the Twins can get a sweet return package for Gray that perhaps undoes some sins of deadlines past, it's hard to imagine the team being better after the trade.
We all know how easy it is to win the AL Central, but trading a guy who has been your best starter feels like waving the white flag. If not signaling that the Twins don't value winning the division this year, it would make them completely unserious contenders in October.
Let's consider the position the Twins are in with Gray. He's in the last year of his contract and possibly his career, so only contending teams would want to acquire him. Any contender that trades for Gray would do so believing he's another piece of a championship puzzle, but if the Twins are contenders wouldn't they also be in that boat of wanting a starter like Gray?
The only difference is Minnesota is driving the boat; the Twins already have Gray and therefore have a valuable piece of a championship puzzle. Looking at it that way, it's nearly impossible to interpret trading Gray as anything other than giving up on the season.
Maybe the Twins still get into the postseason and win the division, but the message will be more that it's a cool coincidence rather than a serious attempt to compete for a World Series. It won't be validating that they still managed to make the playoffs even after trading Gray, it will be frustration that they don't have a guy like him to help them when it counts the most.