The short answer is...there is no short answer. While in graduate school, we kind of glossed over diets as a "fad," defined as an intense but short-lived fashion. There is no real data to say without a doubt that complicated diets work. True data requires experimentation with a control test. During control NO OTHER METHOD can be used. It is often too difficult to have a true control. Also, behavior occurs on an individual level and to test whether or not a diet will work in your particular case it must be tested for the individual not some experimental group. In my opinion if your child is receiving the nutrition he or she needs and is gaining weight (or maintaining) appropriately for their age then trying complicated diets can't hurt. The problem I often see is that diets restricting gluten, casein, and certain dyes often lead to spending lots of $$$ and very limited meal options which your child may not eat - thereby reducing their nutrition.
Now, with that being said, I've seen children whose behavior skyrockets after eating certain foods and I've seen their behavior reduced when certain foods are removed from their diet - with no other changes involved (behavior plans, life events, medication, illness, etc). One thing we did learn in graduate school is that sometimes you have to rule out medical problems first. With that in mind, some people do have actual allergies and sensitivities to certain foods and ingredients in foods. If these allergies and sensitivities make you feel like crap, your behavior is likely to reflect that. All I can recommend is to create your own experiment and find out.
When you decide to try one of those diets, you're probably gonna need a little ABA help! If your child has never been big on fresh fruits and veggies (a likely staple when trying such diets) then you're going to need some reinforcement to help them eat. First of all, try to make it fun to increase the likelihood that they will even try it in the first place. If you have one child (or husband) who is willing to eat, you can reinforce their behavior while the other child observes. In the beginning, you may have to revert to some of their old favorites to use as a reinforcer - "First eat one bite of carrot, then you can have one teddy graham". There are lots of strategies to try to get kids eating non-preferred or new foods. There are also tons of cookbooks out there that are kid friendly and actually make these foods taste good.