Can a 501(c)(3) organization hire a fundraiser to raise money for its affiliated 501(c)(4) advocacy group to hire a lobbyist?
The general rule of thumb is that a 501(c)(3) charitable organization should not transfer money or provide services to its related non-charitable 501(c)(4) organization. The 501(c)(3) is required by the Tax Code to operate “exclusively” for charitable purposes and providing money or services to a non-charitable organization as you describe is not a charitable activity. Of course, “exclusively” in tax jargon does not actually mean exclusively in ordinary English, but it does mean that the charity can jeopardize its exemption if it has any substantial non-charitable activity. The IRS would very closely scrutinize this kind of subsidy.
Charities often create affiliated (c)(4) organizations to do lobbying beyond the limits of what the charity may do itself. From your question it is not entirely clear whether you would employ the fundraiser at the (c)(3) or at the (c)(4) and merely have the (c)(3) pay the cost. In either case, the IRS would be likely to attribute the funds raised for lobbying by the (c)(4) to lobbying by the (c)(3) (at least to the extent of the cost of the fundraiser), which could also jeopardize the (c)(3)’s exemption. (See Ready Reference Page: “Lobbying Rules Create Opportunity for Charities”)
Therefore, while the answer to your question is not absolutely no, it would raise serious risks that could — and should — be avoided by a scrupulous prohibition on transfers from the (c)(3) to the (c)(4). The (c)(4) can give to the (c)(3), but nothing should flow the other way.