It's really hard to answer - what is romantic is an individual definition unless you are going for the quintessential flowers, chocolates, candlelit dinner. But I find those things to be very impersonal choices unless it's something that means something to you like a favourite restaurant or where you had your first date. But to me, the most romantic thing you can do for a partner is to do, plan or participate in those things that makes them the happiest which can even include going back to the same show more times than you can stand smiling while gritting your teeth because that's what he likes to do even though you'd like to poke your eardrums out with a knitting needle (for example). It's less a thing that other people define as romantic and more about showing how you appreciate and understand your partner and what makes them happy or shows thoughtfulness and that you heard what they said.
That said, if you're looking for places and things to do where it's less chaotic and quieter so that you can spend time focusing on each other, there are a lot of different restaurants that are less appealing to children (so they are quieter), tables are farther apart so you don't feel like you're talking to your table neighbours, areas or times in the pools that are not as loud or busy. If you're in a room with a balcony, have a simple late lunch/very early dinner, and then pick up some cheese, crackers, wine from one of the resort stores and have it on your balcony after dark and just chill. Even better if you're at a resort where you can partially face the fireworks at whatever park is nearest and relax with just the two of you. This is one of our favourite things to do.
DH and I did our first trip to WDW after we'd been dating for 6 months so just being in the Disney bubble is romantic to us. Every time we go, it brings back those memories and feelings - and that was 33 years ago. Given that this a dream date for both of you, you're already doing something romantic - sharing something you both love. But maybe you could save the money on the carriage ride and flowers and put it towards something more lasting that will prompt those memories over a longer period of time. Something like Disney jewelry - a necklace with Mickey or Minnie ears (I have the Mickey ears in rose gold and it reminds me of our trips every time I wear it) or do the pick-a-pearl at the Japanese pavilion and have it put into a Cinderella carriage cage.