I share Eric’s concern that such small doses of Levemir may not be covering all 24 hours.
It may be that regular strength Lantus was too difficult to administer in such a small dose, but perhaps that could be solved through dilution, so that instead of trying to give 0.6 units twice a day, you are giving 6 units consisting of 0.6 units of Lantus plus 5.4 units of diluent. There are several threads about diluting insulin on this site. One is Very careful insulin dilution instructions
I share Eric’s concern about a low carb diet for a young child, unless your physician has specifically recommended it. Carbs are good food: they burn clean, are tasty, and suppress ketones.
Learning to dose for carbs is a skill; eating LCHP is a non-mainstream way of managing diabetes that can improve blood glucose metrics without learning to dose insulin well. But the ADA Standards of Care 2023 says clearly on page S74, that very-low-carbohydrate eating plans are not recommended for children. (The 2023 standards of care pdf is available from a link in the left column of the page at Volume 46 Issue Supplement_1 | Diabetes Care | American Diabetes Association )