Campaign funds are those used to promote the election of a candidate. It makes no difference whether the funds came from the candidate's campaign chest, a contribution, or from the candidates personal account. They are all campaign expenses and have to be reported.
If a 3rd party pays money to promote the election of a candidate, it is a campaign expense and must be reported. Who reports it depends on who ultimately put up the money. In other words if Pecker pays hush money, and Cohen arranges reimbursement from a shell corporation that he funds and has agreement with Trump to be reimbursed after the election, then it is Trump that reports the payment as a campaign expense even thou it is after the election. Of course Trump did not report it because the whole purpose of the scheme was hide Trump's part in the hush money payments.
You would give the lawyer a check for $10,000 and record it in your business records as a $7,000 legal expense and 3,000 as an expense that describes the disbursement. If your relationship with the lawyer was that of a contractor, then you would issue a 1099 for $10,000. If you have no agreement with the lawyer, then there would be no need for a 1099.