Well, the Trek 1100 is a road bike and you are going to be limited in tire size. The trail you are going on is packed crushed limestone and not paved so I don't think that bike is going to work out well. I would ride either a hardtail MTB with 2.0/2.1 semi slick tires, or a comfort bike with 32c or so tires.

My experience on shorter stems on road bikes to make up for too long a top tube (bike too big) will make the handling more twitchy, which would not be a + on a crushed limestone trail.

So either rent or buy a comfort/city/touring bike with tires in the neighborhood of 32c. $120 bucks is a good chunk of a nice city bike and you can also use it on those rides with your son that you mentioned.

The Trek 1100 is a really nice bike, a classic, but about as good a choice for that trail as your Madone.