I was about to say, no, you can't do that if you're picking up from the Car Care center at Disney, because there is no
Alamo or National agency at the port (that I know of). However, the second link suggested above suggests that you can drop off a National rental at
Enterprise near the port. I suggest you look into that and make sure you can do that.
I know many rent from MCO and drop at the port (and reverse), and I'd also encourage you to check that out. There's a chance it could be less expensive, even though more days.
(Actually, what we found less expensive than all of this was just renting MCO to MCO and pay for parking at the port. One other benefit of doing this is we were able to stash things in the car, such as dirty laundry, from the WDW portion of our trip so we didn't have to bring it all on the ship, where space can be an issue)