Why do low carb diets cause bad breath? Is there a way to fix this without loading on carbs?

Yes. Low carb diets are rich in proteins. Odor producing bacteria in your mouth love proteins , so they multiply... Sometimes rinse with diluted peroxide (1:1) may help. If not - find a dentist who can do "full mouth detox" we do it exclusively in our breath center in calabasas, ca.
Ketosis. Your bad breath is probably the exhalation of he tone bodies aka ketone breath. Try chewing sugar free gum.
Eat "odor-free" food. A person eating a low-carb diet should be able to avoid food-related bad breath if he chooses foods that are not likely to cause odor. Avoid fishy foods, onions, garlic, scallions, fermented foods, coffee, etc. Fatty foods may worsen stomach reflux, and send odors up to the mouth. Excess fiber or other "undigestibles" can lead to gas and odors coming out the other end.