I think your DP's right.
I had absolutely no hesitation about running while pregnant, assuming that, on the day, I felt good. I did a zombie run last spring myself (not while pregnant). I would not take on a "safe but challenging assault course" (I googled the event) while pregnant. I think you'd end up with pulled muscles, and put yourself and your partner through a lot of unnecessary stress about other incidental crap you'd encounter on the course, stuff that's unlikely to seriously harm you or your baby, but which is not going to be good for anyone's peace of mind. I also suspect that if you told the staff that you were pregnant on race day, they would try to tell you not to run the course, to protect themselves from liability.
The race organization does not provide course maps or obstacle descriptions, which is standard practice, and even part of the appeal, for this kind of event. However, I don't feel comfortable taking on that level of uncertainty about what I'm going to be doing on the course while pregnant. I want to be able to train and make sure that I feel safe with the activity.
Rope nets seem to be a thing for this event - they show up in a lot of pictures. Climbing unstable things (like a rope net with a bunch of other people on it) is a great way to pull a muscle if your hip and ankle are suddenly stretchier then usual, as they are when you're pregnant.
The website mentions slides, and so do you. The pictures on the website include a bunch of tube slides that end (sensibly, really) in a big pool of water. The pictures show only one guy in the water. This almost never happens during the actual run. Because of the multiple possible start points, the fact that people slide down at different rates of speed, and the need to keep runners moving through the course, you're highly likely to get slammed by someone behind you shortly after you hit the water. The far side of that pit is built to be easy to climb, not soft to slam into.
Obstacle labeling for these events is not always what I'd want it to be. For example, my zombie run included an unlabeled electric fence - a couple of my friends got zapped and briefly passed out. Again, not okay by me while pregnant.
Take a pass and do this a few years down the road.