It is highly unlikely that reducing carbohydrate will cause tooth pain. In fact, the vast majority of people on a ketogenic diet experience the opposite. Tooth pain goes away once sugar, starches, and grains are eliminated from the diet.
If your ketogenic diet is lacking in minerals, I would suggest switching to whole, real foods; that is, meats and vegetables. You will need to increase salt intake a bit, for reasons explained many times on these forums (the search function works well, here, by the way), and that will help keep your other electrolytes balanced. You should also be drinking to thirst.
Ketone bodies generally remain in the bloodstream, except when excreted in urine or breath. If you are indeed breathing through your mouth enough of the time to make ketones a problem for your teeth, that is a symptom that needs to be looked at, as it indicates a non-keto-related problem.
Gums do not recede from lack of insulin. Populations that eat diets that keep them generally in ketosis have a vanishingly small rate of gum disease or dental caries. In fact, an increase in the rate of cavities in a population, almost invariably associated with an increase in carbohydrate intake, is one of the first signs that the diet is changing for the worse, as Dr. Weston A. Price demonstrated, years ago.
As for oxalate dumping when plant-based food is reduced, the best course of action, as recommended by Dr. Annette Bosworth, is to ride it out. It’s like the Candida die-offs experienced by people with systemic yeast infections who go keto: it may be unpleasant, but you are better off getting through the unpleasantness because the increase in health afterward is worth it.
Your best course is to look for possible sinus problems and deal with them, since that is the most likely factor for reducing tooth pain.