We took our youngest for his first trip when he was 9-months old... what a great trip!
For the plane ride, I did not get him his own seat.. I knew our flights would work well with his napping.. and I just held onto him the whole time. It was kind of tough on my arms... but a few pillows worked well... and DH helped out when he woke-up. I did try to keep him awake for take-off and landing... to help his ears pop - I fed him during this time. When he was awake, he played with a few toys. I remember he kept dropping them - so perhaps toys that you can tie to one of those straps...(I forget their name... they have them in CVS and Walgreens... they are for strapping on bottles or sippy cups onto stroller/car seat)
Yes, bring your own stroller... DS took many naps in it and was very comfortable... he slept everywhere and it would have been difficult to wake him to turn in a Disney stroller when done at the parks. One evening he fell asleep in our stroller right before dinner in Norway... I was able to bring him into the restaurant, in our stroller, and he slept through dinner. It was wonderful, relaxing, so nice! If he was in a big Disney stroller, I don't know what we would have done! He slept on the monorail... The funny part is that he really is a
horrible napper.. but when he got tired enough, he slept and we took him along! I found a clip-on battery fan that I used on his stroller. This was a great help. I turned it on whenever he took a nap. I also found a full sunshade for the stroller. This was helpful when walking, many times we were facing the sun and my stroller couldn't cover him completely (I have a Maclaren). Both items were really important during our trip
I didn't see a date of your trip, but if it will be hot - you will need to keep him hydrated (as the other poster mentioned) We fed DS ice chips sometimes.. and even resorted to (watered down) slushy's once or twice. I gave up feeling guilty about some sugar - because I knew he was getting extra fluids.. it was May and it was HOT..!!
We brought along a baby carrier that was meant for swimming.. by Body Glove.. It was very handy at the pools.. and it was really lightweight to put in the bottom of the stroller.. (but the neoprene was hot..) Even so, I still like the idea. When we went on some rides (like the safari) I would leave the stroller and throw on the carrier (I could do this myself - very important) and plop him in. He was able to be close to me, sweat a little

and if I wanted to take the carrier off it was easy too.
Oh, I've never stayed at BCV, so I have no specific tips for that resort! But I hope this helps! Have a wonderful time!!