I am LADA as of 2 years ago right before my 50th birthday. Have you had a c lettuce or GAD test. That will help determine what type of meds you need. Some people only need to temporarily go on insulin because they're sugars are too high and the pancreas can't keep up or catch up. Some like myself are less fortunate and need to be on insulin forever but I am much healthier for it.
Have you lost weight recently without trying?
As far as LCHF, if you have fat on your body you don't need to eat a ton of fat because if you are low carb your body will burn its own fat. For me , the type of fat I eat makes a huge difference. Saturated and dairy fat are the worst. I do well with avocado, mayo, olive oil and a few nuts and seeds. The rest of my diet is above ground veggies and moderate amounts of protein. If we eat high quality foods we don't need nearly as much because there's no junk to process and the nutrients are more abundant. I don't know anything about the meds you are taking but LCHF does not need to be LCHF gob the fat. Plant fats are easier to process and you only need moderate amounts and they need to be healthy fats, not processed seed oils, trams fats etc.
As far as constipation, avocado is my laxative. Always had issues but not anymore. I eat one a day divided into three meals. Great healthy fat, the right kind of fiber.
I'm not clear on why you can't eat eggs? BF doesn't have to be eggs and bacon. It can be chicken, turkey fish etc. I often eat half an avocado and celery or a large piece of lettuce with mustard mayo and a slice of deli turkey with some avocado. Lunch and dinner are basically the same. A lettuce wrap or salad with some protein, avocado and a dressing/ sauce of choice. Either olive oil or a mayo mixed with either hot sauce, mustard or horseradish. No need to get fancy.