Ok. Here are all the examples from the Cambridge Advanced Learner's Dictionary and the American Heritage Dictionary for turn down. As you can see, in none of them is the noun object to be found between turn and down, it's always after. When the object is a pronoun (it/him etc.), it is placed in between - actually it has to, this is a grammatical rule, as I believe you know. If the object is a noun, the rule says that you can either put it before or after the adverb (here: down). So both of your examples are technically correct, but I'll insist that the word order in your first example is the preferred choice. All the more so, since you're using not just one but three words (all four offers).
He offered her a trip to Australia but she turned it/him down.
He turned down the job because it involved too much travelling.
She turned down the opportunity to work in Paris.
He asked her to marry him, but she turned him down.
I heard that he turned down a knighthood.
He turned down the chance to have his own exhibition.
Would you turn down the chance to meet Nelson Mandela?
We politely turned down the invitation.