To the standard Suburban rental....
If you are a gold member, and have added all your requisite information to your profile (payment card, license, passport), you can usually skip the main office, head straight to your car, and you can add the additional driver at the exit gate. Depending on some other factors, especially what CC you sign up with, I could help you status match to one of the elite Hertz levels - PM me for details. It might be a good idea that you update said online Hertz profile and do a new member document check at your local hertz in FR, that way your profile has the "green light" and you can utilize the Gold canopy at LAX. Otherwise, first rental for a new member they usually want to see your documents at check in.
Their will be a nominal daily fee for each add on driver, except for same household spouses and business associates. However, their are ways of getting around this (status or select CDP codes, do some searches in this forum's threads). The exit booth will check your license and payment card, and perhaps your passport. Note, I have only used either personal or work expense credit cards for my reservations, can't speak to a debit card use. When I have added on the extra driver, they just check the license and we are on our way, no extra payment card check. These were for domestic standard rentals on a US Drivers License however. The one international rental we did in Ireland was at a local hertz and was all desk check in w/ each of our Passports, US DL, and my CC.
Can't speak to a true reserved adrenaline car. Got upgraded to one, Challenger RT, and all that was required was a drivers license check at the booth in the garage. No extra steps for extra credit cards or anything like that. Not sure if Hertz PC status and being a frequent renter in their system came into play there.
Not that I am trying to sway you away from Hertz, but looking at National or even Sixt might be an option - they are a bit more open to the extra driver situation than Hertz.
PM me with how to get status matched into any of the 3 brands above - that will unlock some more options for you and your group.