How can a husband tell his wife that she has bad breath without embarrassing her or hurting her feelings, especially given that religion is a facilitator?

Estimating a wife's feelings is a sign of good character, and your question about how to inform her of her bad breath without hurting her feelings is an evidence of that. Religiously, there is no harm in informing her. You can guide her to use miswak without mentioning the odor, and if the odor persists, you can use subtle hints, such as asking about an illness or a sore throat, or if she has eaten garlic, to make her aware. If she doesn't become aware, then direct confrontation is obligatory.