Let’s start with the concept that you don’t “own” either the (c)(3) charity or the (c)(4) social welfare organization.  You may control both, but you don’t “own” either. 

Second, if you had two organizations with the same name, you would still have two separate organizations with two separate sets of requirements for tax-exempt status, not a single organization with dual exemption status with the IRS.  The IRS would recognize the original (c)(4) as a social welfare organization.  It would recognize the newly named charity as a charity with a name that just happens to have the same name as the (c)(4).  Since they each have a separate Employer Identification Number, the IRS will be able to keep track of the differences.  You would have to keep the financial books and records separately, file separate tax returns, and meet the requirements for each type of organization.

What you would have is monumental confusion.  How would your donors know whether they were dealing with the charity or the (c)(4)?  How would they know whether they could claim a charitable contribution deduction or whether the gift went to the (c)(4) lobbying effort?  How would the people you were lobbying for changes in legislation know whether the charity was doing “too much” lobbying if each organization had the same name?  It is not a good idea to have two organizations with the same name dealing with two types of business.

Depending on what you actually plan to do, your premise itself may be flawed.  Charities are allowed to lobby.  They just can’t do too much.  (See Ready Reference Page: “Lobbying Rules Create Opportunities for Charities.”)  Social welfare organizations can do charitable things.  It may be possible to meet both missions in a single (c)(3) organization. 

But if not, the more traditional way to deal with the issue is to have related organizations with similar, but different, names.  You could have the Do Good Organization as the (c)(4) social welfare group to do huge amounts of lobbying and the Do Good Foundation as the (c)(3) charity to conduct charitable activity.  And by looking at the name of the entity people were dealing with, they could tell which one they were supporting.  It would make a lot more sense on an on-going basis to have separate names if you have separate activities.