There are a couple of issues here. Firstly contractor vs employee. Having recently gone through this exercise, unless the appointment is literally for just a few weeks, most professional locums prefer to be employed, i.e. via PAYE as this makes a lot of functions easier such as access to the bank etc. If the appointment is to cover a short period (no more than a month or so), then contractor status is acceptable (even to NALC). Employed status implies payment as per agreed salary scales, even if the level on that scale is different from the proposed permanent clerk whereas contractor status suggests a contract rate which may not reflect the rate applicable to any of the salary scales. In generic terms, in my view discussions about budgets and contractor vs employee wouldn't reflect personal details so why "confidential"? And just to reiterate, where the discussion is confidential, the decision is not.
Once the first decision has been taken, i.e. employee vs contractor, then the selection process follows your usual procedures. If an employee, then your usual recruitment procedures would prevail. If a contractor, then your usual contract selection process would be appropriate. In both cases, the point where you discuss an individual, then it becomes confidential but as I've said, the result is not confidential, albeit you should be a little wary of naming an individual (as oppose to a company, even if a contractor) followed by their remuneration in the minutes as this would breach GDPR rules. And for the record, confidential does not mean excluding councillors, but excluding the public from the meeting room.
Payments can be reported in the same way you report any other payment. If a contractor, it's on the payment schedule as usual. if an employee, it is listed in the same way as you would normally list salary payments. I agree there are differences of how this latter payment should be shown and for a small council whilst its a bit of a nonsense to withhold the information as it's going to be shown clearly on the AGAR at the year end anyway, it is not a payment that can be withheld as such so not one for "approval" by council.