Your bike is fine. I have ridden parts of this trail with an old Miyata touring bike(tires32s) I would not use a full suspension mnt bike as it is too heavy and overkill for this type of bike path. I actually feel that a race bike w/ 23's would be fine. Very nice trail with beautiful scenery and towns along the way.
Not a fan of the trailer but if that what you have, use it. From what I have ridden, a shock is not necessary on that trail. I really like some of the "Dutch Cargo " bikes. Pricey for sure but will give the kids a comfortable place with room for gear. Plus it is easier for you to keep an eye and have conversation while you ride. Not ultralight for sure but enjoyable for the little ones. You won't win the Tour de France with it:) I wouldn't buy it for 1 trip but if cycling becomes an important part of your lives, consider it. I wish I had know about it when my kids were young. Here's a pix