Voragh already discussed the part about {-Daq} being unnecessary with {'el}. As for causatives with two objects, there are a few examples where the thing being caused to do something (in this case, the dog) is marked with {-vaD}. The thing being entered (or the object of the verb, generally speaking) doesn't change its role in the sentence. The pattern seems to be that if X causes Y to do (verb) to Z, the sentence is:

Y-vaD Z verb-moH X.

So the sentence you're looking for is {dogvaD pa' lu'elmoH}.