It's a very interesting question and we applaud the research you've done so far (no pun intended).

First we are going to make a few assumptions.  First, that when you write "converted to a permanent employee thereafter" you meant the individual is hired by the prime contractor and becomes a prime contractor employee.  Second, that the tasks these subcontractor and prime contractor personnel are/will be performing have already been vetted as not being prohibited by: FAR subpart 7.5 and DFARS subpart 207.5 (inherently governmental functions) and DFARS part 237 (e.g. firefighting, see DFARS 237.102-70).

Outside of that, we see no prohibition on the prime being allowed to use a staffing contractor to meet surge staffing support requirements.

Does the prime need your authority to do this?  If they don't have an approved purchasing system, follow the procedures at FAR 52.244-2(c).  If they do (which we suspect), you only need to Follow FAR 52.244-2 if this type of subcontract (e.g. staffing surge support) is listed in the fill in at FAR 52.244-2(d).  I think paying the prime's voucher of the ODCs under 52.216-7 constitutes enough approval.

How would the subcontractor (staffing firm-contractor) be paid?  This mostly depends on the prime contractor's approved accounting practices.  For simplicity, we are going to assume your prime contractor has multiple subcontractors and/or vendors and that they also have multiple government prime contracts and not just yours.  Until they become prime contractor employees, all of the direct costs the prime incurs from the subcontractor in support of your contract would be chargeable to the prime contract as an ODC.  The administrative costs the prime incurs in things like awarding the subcontract, etc. would most likely end up in the same O/H pool the prime incurs for the administrative oversight of all their other subcontractors.