It really does depend on the days you want to rent for, as well as how long you will rent for. I recommend looking at the Costco travel site, as I just found a fairly good deal for my car rental, and you don't need to be a member to look. You have some limitations that you stated above, as you will need to pick up from the Disney Car Care center, which narrows down your choices a LOT (to only one car rental brand; Alamo). If you are willing to take an Uber from your resort to one of the other local car rental places (you can specify a van), or use a resort near Disney Springs, that you could take a bus to, it could save you a lot of money. If it was me I would check my bags with bell services, take a bus to Disney Springs with my family, and walk over to the Hilton so both adults could sign onto the car rental/get a free driver. If your significant other will not be driving, head on over yourself, drive back and pick them all up with the luggage at the hotel.
For example, for a weeks' worth of days in August, I found, on the Costco site, that you could rent from the Disney Car Care Center and drop off at the Naples airport a
minivan for a week for $741. Or, for the same dates, and taking an Uber over to the Hilton Hotel near Disney Springs, (or even a bus from your resort to Disney Springs Marketplace - it is an 8 minute walk to the Hilton from the Earl of Sandwich per Google) you could rent a
minivan from Avis for $450.45. You would save a lot more than the cost of a Costco membership. You could also make the reservation now, keep checking for better rates, and then cancel.
Alamo rates on their own site (they are at the Disney Car Care Center) are higher. Costco rates include a second driver. Rented through the Alamo site, Alamo charges avg. $12 a day for a 2nd driver, per their site.
If you need the car for just 24 hours, dropping off at the Naples Municipal Airport, I was finding
rates in August for a minivan for $84 (Hilton/Avis) or $89 (Disney Car Care). That doesn't seem bad at all for dropping off at another location and having a second driver.
Prices are all over the place. I recommend you think about any affiliations that you have, or would be willing to have if it would save you big $$$. A $55 on a Costco membership could save you big $$$ on a weekly rental, but not so much on a daily rental. Maybe you have
AAA, or have another membership. They all have their own "affiliate codes." There are a couple of Hertz locations you could Uber to on the Disney property, and Hertz often has stackable deals/coupons on their site.