DART nudged it's target about 2.7mm/sec. To ensure a miss you need a sideways deflection of up to 4000 km (depends on the geometry.) To generate a miss by Dimorphos assuming the worst case you're looking at 1,481,481,481 seconds of lead time. That's about 47 years. Against the dinosaur killer you're looking at a lead time far beyond our ability to predict orbits.
Note, also, that if you're throwing kinetic stuff at your target you get little choice in how it hits, you very well might not be able to shove it to the side.
Despite the refusal to consider the option there really is only one choice against a major threat: Orion. We know it works (it has been flight tested with conventional charges, it has been tested with single impulses with nuclear charges), while there are some serious question marks about whether you could actually build an Orion capable of launching form Earth (how do you keep the pusher plate from getting too hot??) that does not apply if you're using it against an asteroid. You don't care what shape the asteroid ends up in, just that it get out of the way. Your only limit is that you don't want to make any given shove big enough to risk breaking the asteroid.
By a crude approximation you can easily get 10% of the bomb energy to show up as impulse, this can be increased by putting the bomb closer (remember, you don't care about damage short of destruction) and some still-classified data says that it's possible to build a shaped charge that directs the majority of it's energy at the asteroid. Obviously, exact numbers are not known to us mere mortals.