Here's some reasoning behind why Paleo might work for IBS/bowel problems, and why it might not work, depending on how you implemented it. Note: I have Crohn's (inflammatory/autoimmune bowel disorder) and a low-starch, low-fiber Paleo diet seems to be working for me.
First, we need to talk possible causes. The overarching theory that ties all of these together in the neatest package is fairly simple: your diet is causing a pathological mix of gut flora to arise, and your genetics then determine your response to that pathological state. Maybe you only get IBS, maybe you get Crohn's, maybe you get Ulcerative Colities, but underlying all of this is some level of dysfunction in your gut ecosystem. There is some experimental backing for this hypothesis: google "Ebringer starch" for some information about a researcher who discovered linkages between a certain bacterium and Crohn's, and read this from Kurt Harris about a guy who has discovered linkages between malabsorbed carbs and IBS.
So why might Paleo work? Well, if your take on Paleo has lead you to a diet heavy in meat, low in starch and low in fiber, at least in theory, you should now be eating in such a way that the bacterial actions that underpinned your disorder should be significantly reduced. In the case of IBS and FODMAPs, if you are avoiding excess fructose and various hard to digest poly-saccharides, then you should be reducing the amount of small bowel/colon fermentation, and so reducing the triggers for the disorder.
And that should give you plenty of hints as to why it might not work. If your Paleo take has you eating a lot of fruit, vegetables like onions and asparagus, then you might be simply swapping sources for FODMAPs.
If you did google Ebringer, then you might have come across the London AS Low Starch diet. You will notice that this focus on starch restriction, which also covers FODMAPs but does not focus on them. If you interpret "low-starch" as "lots of veggies", you may fail. If you interpret "low-starch" as "lots of meat", then you'll probably get it right. If my n=1 experiment has any weight, it seems that a VLC diet works. It may work by accident; that is to say, it may work because it restricts the problem foods as well as harmless ones. I don't know.
I can't say whether mere low FODMAP would work, and indeed there are other researchers out there who will tell you that you need to eat more inulin and fiber so that you can grow gut bacteria. My own experience would disagree with that interpretation. Fiber and fermentable carbs give me gas and cramps. Oddly, when I challenged myself with a pile of mashed potatoes, I felt fine - but eat some brussel sprouts, and I'm farting like crazy. So watch out for any food that might not be readily digested and absorbed.
One final note: My own success only started to appear after three months of digestive turbulence. Do not waver on day five. Set a goal for yourself to eat this way for six months regardless of how things go.
I hope you get it under control!