Low carb wasn’t designed to put people into nutritional ketosis (although many of you might actually be in ketosis when you’re eating a low carb diet).  Low carb was mainly designed to restrict your carb intake.  Keto, on the other hand, is designed to put you into nutritional ketosis.  And it just happens that in order to get into nutritional ketosis, you decrease your carb intake and increase your fat intake.
There are now few years since I started educating patients with overweight, obesity or signs of the metabolic syndrome about the possible health benefits of LCHF. I have seen many of them having a great success with this lifestyle. Most have managed to improve their general health and modify their cardiovascular risk factors in a positive way. By this, I’m not saying that other methods don’t work, but I can confirm that LCHF very often works in real life.
Jan – It takes a while for your body to become “fat adapted”. Every person is different, but my guess is that your body is probably still adjusting. I’m not a medical professional so please don’t take this as medical advice in any way. But I also had this issue and it turned out that I was going a little heavy on the coconut fat, which can be a stomach irritant in large doses. It’s okay to back off a bit to give your body that adjustment time. Increasing fats slowly over time can make things a lot easier. I know it’s tempting to dive in head first, but your body is telling you something, so you should definitely pay attention. Google “Fat adapted” for more info. I don’t know a lot about this. All I can share is my personal experience. For me, it was too much fat too quick as well as the type of fat I was using. It may be something totally different for you, so you’ll need to experiment to see what’s causing it. Keep in mind that a Keto eating plan is a tremendous change to the body from a mainstream eating plan. 6 weeks is not a lot of time. But if it persists, it’s probably best to keep in contact with your doctor on a regular basis. Kudos to your doctor for being open to this way of eating!
Another thing to (not) consider is the GI index of carbohydrate. Regarding this, you should be aware that the glycemic index of foods has no relation to low-carb eating. Although many people associate ‘high GI’ with bad and ‘low GI’ with good, all this means is that the body digests some carbs slower than others. No matter the speed, they are still all digested and contribute to the carbohydrate total.
Other things that are fantastic for milk supply are fenugreek capsules (start with 3 caps three times a day for about a week, then reduce to one, three times a day as your supply kicks in). Brewers yeast or nutritional yeast are extra helpful, as is coconut oil! My little bub gained lots of weight and my milk suply was great after doing these things. (My previous two children I had a lot of trouble with supply). Small portions (again, about a 1/4 cup serving to limit carbs) of quinoa once a day are also helpful when breastfeeding.
Thank you for writing this post, it’s very helpful! I started low carbing a while back (probably more than a year ago) and ended up backsliding, as can often happen when trying to drastically change your eating habits. On Friday I had my annual physical and my doctor ordered blood work for glucose and insulin because previous results indicate I’m insulin resistant. The results from Friday indicate I still am, so now I’m a little worried and going back to low carb. One phone app (and website) I’ve found to be really helpful is Fat Secret. You can choose exactly which macro nutrients you want to see and it also shows percent of fat, carbs, and protein in a pie chart that I find helps me wrap my head around what I’ve consumed. It also helps that the app and website don’t seem to promote one eating plan over another.
Please share more info., e.g., starting weight, amount of time you exercise and what intensity; how many carbs/day and do you count net carbs? I started keto in April and did well (lost 12 lbs.), except I had some mood issues at first. I really struggle to stay in keto and I just don’t think I can long term. I can definitely stay around 50-75 carbs but not 20. I have maintained what I lost, but I can’t seem to lose any more. I take 1,000 mg. Metformin daily, weigh 163 and am 5’4”. I really want to lose at least 10-15 more pounds. Maybe I can if I do what you’re doing.
Under normal circumstances our body uses glucose from carbohydrate foods for energy. In the absence of glucose a process called ketosis occurs. This is a state in which the body burns fats instead of carbohydrates as its main fuel source. When we don’t eat carbs, the liver breaks down fat stores to produce energy. This energy is in the form of (and also creates) molecules called ‘ketones’. 
Great Q. While the answer isn't totally black and white, Zeitlin says that for most healthy people, if you're trying to lose weight, a low-carb diet would be the better fit because it doesn't aim to put your body into an unnatural state (ahem, ketosis). Instead, by cutting back on foods that are high in sugar and starches, you're able to drop weight and still maintain a healthy form of energy for your brain and body to run on.