ON a low carb diet, you are replacing the missing carbs which previously were your source of energy, with extra fat and protein instead. Once it's got used to the changeover, your body will quite happily run on fat.
In general, to get into this fat burning state (ketosis) you will need to reduce your carbs to under 50 grams a day to start with. The Atkins diet as far as I can remember suggests 20-25 grams of carbs a day for 2-3 weeks to kick-start this change. After this you should try to "eat to your meter" to see what foods may be spiking your glucose levels after meals, and watching the scales to see if you are continuing to lose weight or not.
If you need/want to to both lose weight and reduce your BG levels, then you do need to a certain extent need to watch both fat and calorie intake to ensure that your body will use some of its stored fat, as well as the fat you eat, for its energy, so "As much fat as you need in moderation" is probably a better idea than an unqualified "As much fat as you want". The proportions of carbs, fats and proteins will vary with everyone's individual situation, as we're all different.
There is no particular virtue in keeping to a very low carb diet if your blood and body can tolerate slightly higher levels. So. use your meter to check on blood glucose levels and which carbs cause after meal spikes, and use your scales to monitor your weight, and adjust your fats and carbs accordingly.
Robbity