As far as strollers go, I think the two biggest questions you should ask yourself is a)will you want the stroller in the airport and b)how easy does your stroller fold?
I've done all three different ways. Brought my own, rented from offsite, and used a park one. I wouldn't recommend the park ones since Squish is still so young. They are plastic and hard as a rock.
If you can't fold your stroller with one hand, it might be beneficial to renting offsite, only because schlepping the items on and off buses with a bag and a baby can become cumbersome if said item is difficult to fold/unfold. However! The downside to the strollers that fold one-handed (I'm speaking specifically of City Minis) are the storage pockets aren't very big, so if you think you'll want space to store bags, waters, etc and your personal stroller is great with that, then that is something to consider, as well.
Naps back at the resort are always great, simply because you know the baby is getting solid and restful sleep. I would just say, though, that your little might get overstimulated in the parks and fall asleep sooner than you expect. The first day of my son's first Disney trip, he fell asleep at eleven when his usual naptime was after lunchtime. It's just a lot for their little eyes to take in! Or it could go the opposite direction, too, and you get Squish back to the room and she struggles to fall asleep.
Good luck with the baby-proofing! Some of those latches are difficult for adults to get through.