Hi and welcome.
1. No, it's not unsafe as long as you eat enough. There are plenty of people, including children, who use a ketogenic (LCHF) diet to treat seizure disorders and other brain disorders. I haven't heard of dangerously-low body weight being an issue. Same as folks in maintenance who've eaten this way to lose weight, and continue eating fewer than 20 g carbs per day in maintenance. You won't lose too much weight unless you ignore your appetite.
2. Seeing as the opposite problem is occurring here, shouldn't you try stopping the laxative?
Other than that, some people say they get diarrhea or constipation initially, but I'd say that would be true for a lot of people any time there's a drastic change in diet. It takes time for the body to adapt to new foods in different quantities.
If you were getting 70-90 g fat before, it shouldn't be a huge increase now that you're eating LCHF. To me, the "high" in high fat is mostly as a comparison of percentage of calories. When you slash the carbs, the fat percentage goes way up even if you didn't actually increase the quantity of fat.
Did you suddenly increase your intake of a particular food such as cheese, nuts or coconut oil? I'd still be looking at the laxatives as an obvious cause of the diarrhea here, though. Wouldn't it be nice if you could cut down or stop because your gut is healing?
I'd suggest going to the doctor on this if it continues, but be careful how you describe your new diet. I'd avoid calling it low-carb and/or high-fat. Say you've cut sugar, junk food and processed foods, or something like that.
Best of luck in finding a solution.