There's a large enough difference between 9.7 billion and 3,628,800 that there's no contest, even if the permutation engine was slow. And there are optimized ones around, like Algorithm::FastPermute.

But check out the hawtin solution below, which generates a combination one letter at a time, eliminating large batches of cases with early constraint testing. And all the constraint checks are simple concatenations and arithmetic.

It's not a completely general solution as it stands, but it could be the basis for one, by using generated code.

Update: It could also be generalized fairly easily by a looping program that pushed the current state onto a stack, or by a recursive program.

Update 2: I just noticed that your idea to test only combinations that sum to 45 means you could count by 9's. Only multiples of 9 have a digital sum that is a multiple of 9. That cuts your number from 9 billion to about 1 billion... still too many to be worth doing it that way, IMHO.