Why isn't the dogma of predestination a prescription for indifference?
It's a tough question, because predestination is a mystery of the Divine Will, but various ways of addressing it have been attempted.
There's the pragmatic, "ours is not to reason why" approach, which points out that Scripture tells us to choose good and avoid evil if we want to be saved, not if we know we are going to be saved. We follow Christ's commandments because following His commandments is our job; worrying about how what we do meshes with Divine predestination is above our pay grade.
This suggests the "if it quacks like a duck" approach, which looks at something like Fr. Garrigou-Lagrange's "signs of predestination" and reasons, "If you bring about these signs in your life, you're doing a good job keeping up your end of the Covenant." Jesus did, after all, make various promises He will certainly fulfill, but these promises all require us to do something. (This is not salvation by works, but salvation following upon faith, which can be shown to exist through works.)
But I think there's also a metaphysical reason why predestination doesn't imply indifference.
Scripture tells us God's ways are not our ways, but do we see what this really means? It's not merely that God's ways and our ways are disjoint subsets of some set of all possible ways -- as we might say "Russian ways are not Algerian ways" -- but that God's ways and our ways cannot possibly be classified in the same set. "God's thoughts are not our thoughts" doesn't mean that, of all possible thoughts, God thinks some of them and we think different ones, like "My thoughts are not Bishop Griswold's thoughts," or even, "My thoughts are not a fly's thoughts." It means that what we call "God's thoughts" are not of the same order of being as our thoughts, that though we can speak of God's thoughts analogically, the differences between His and our's are greater than the similarities.
This means that predestination and human freedom are not contradictory -- in fact, they can't be contradictory, because they are in no way comparable. It's like the way a musical note can't contradict a tree; if anything, saying, "Middle C contradicts that oak tree," makes more sense than saying "predestination contradicts human freedom," because notes and trees are at least both elements of Creation.
Now, we are saved by God's sovereign will and our faith in His Son, but the "and" here is not additive, because God's will and our faith are not commensurate. It's a bit like saying a mother is pleased on Mother's Day by her daughter's cookies and her son's song. The baking adds nothing to the singing, and the singing nothing to the baking, but together they result in the mother's pleasure.
(It's a weak simile, admittedly, since the mother would presumably be pleased with either one by itself, and our faith depends on God's grace in a way the singing does not depend on the baking, but it's the best I can think of right now.)
In short (if it's not too late to be short), predestination doesn't mean God is "doing" something for our salvation, so we don't have to. God's doings are not our doings, as you might say; His will for our salvation operates on a different order of existence than our own, and our salvation depends on operations in both the Divine order and the created order.