I don't juice and haven't tried the diets mentioned (though I have tried similar ones), so my opinion is far from expert, but I'd be cautious experimenting with restrictive diets if you are unhealthily underweight.
One of my doctors suggested I might want to try the FODMAP diet recently, but after looking at it I decided not to - the doctor had been wondering if small bowel bacterial overgrowth was contributing to my symptoms, but it seemed far from clear that that was the case. There is a test to diagnose bacterial overgrowth, but it is not widely available, so the doctor said you can either try the FODMAP diet or try antibiotics, and if things improve, it confirms the diagnosis. Since I'm very underweight and my diet is already restricted because my stoma does not tolerate fibre well, I didn't want to do the diet. I ended up taking masses of antibiotics for another reason, and my bowel symptoms did not improve, so we concluded that bacterial overgrowth was not my problem.
I guess you'd have to consider whether you feel bacterial overgrowth is causing your problems. I recall that it is also claimed that the FODMAP diet can help with Irritable Bowel Syndrome, which made me even less convinced that it could help; since Irritable Bowel Syndrome is used to cover a whole load of different symptoms, and likely given to people who actually have some other bowel condition which just hasn't been diagnosed properly, a diet that is supposed to help with it seems dubious to me.
But ultimately radical diets tend to conflict with attempts to gain weight and attempts to spend less money on food, so you'll have to be careful weighing up the potential benefits and costs.