As Cindy has touched on the age 6 requirement it is , IMHO, a little arbitary.
DC has to lay down some sort of guidelines as to what the physical requirements will be. Age 6 is IMHO a reasonably conservative estimate, most 6 year olds should be OK with this activity but it is their ability and CONFIDENCE in water that is most important. They could just have easily said kids have to be 45 inches to do this and that requirement wouldn't have been a better ( or worse) guideline. There are kids aged 4 or 38 inches would be fine with this experience ( my elder daughter would have coped fine at that age) and there are kids age 10+ and 60 inches that wouldn't be able to do it ( my younger daughter is now 12 and I think she's now only just about mature/confident enough in the water to chance spending that much money on the swim, even though she's been a more than adiquate swimmer for 3 or 4 years. As a parent you know your child's ability better than anyone else. If he's capable of spending 5 minutes in the deep end of a swimming pool without help, he's going to be physically able to complete the swim. If he sight of a VERY large dolphin isn't going to put him off then he's going to be mentally able of enjoying the experience.
IF you don't think he's up to the task of swimming in the deep water, you can have the "one on one" time with the dolphin in the shallows where he can stand. If you think that would be beyond him then there is a chance he could get a little time with the dolphin. Friends of mine went on the swim with one child who wasn't capable of doing the swim, she behaved very well and watched patiently while everyone else enjoyed their time with the dolphins. At the end my friend asked it would be OK for her to pet the dolphin when everyone had finished and the CM very kindly let her have a couple of minutes. She was overjoyed to get the chance. I'm sure that her good behaviour did influence the CM in their kindness and it isn't something you can count on, it depends on both the CM and the dolphin being willing to extend the experience a couple of minutes.
Ultimately it's a lot of money, if you're not sure of his ability or reaction then he shouldn't be put into a position where the experience becomes an unpleasant one.It would be a shame for everyone else's enjoyment being put on hold because you think he will create a scene. There are a lot of activities he can take part in and I'm sure he would have a great time. There are times in life we can't have exactly the same experience as everyone else, due to our own short comings, in this case his age. I hope that he would be mature enough if you sit him down and explain that while there is a lot of fun things to do at DC, he isn't yet old enough or a good enough swimmer to take part in that particular activity.
I would think if given the choice of going and doing everything else other than the dolphin swim, or not going at all, he'll choose to go. Particularly as you're on very short notice you could sound him out as to whether he'll accept not being able to do the swim, if you think he's going to be OK you can try and book it. If not, blame DC for being fully booked. He doesn't need know that you decided it was better not to take him, and neither does the rest of the family, who may be a little resentful. JMHO.
I think it is a requirement for one of your group to do the swim.