The term "carrier oils" typically refers to the same oils that we use to make soap, or to dilute EOs. Because they are oils
, they won't disperse into water without a solubilizer, like PS80 or dishsoap. However, neither PS80 nor dishsoap (even your own liquid soap made with KOH instead of NaOH) would be moisturizing.
It's hard to find one product that will both disperse and moisturize. One exception is turkey red castor oil, aka sulfonated castor oil. I've seen it used in a few bath bomb recipes. You can find it here on Amazon
, where they have this description:
Castor Turkey Red is an aqueous solution of sulfated castor oil. It is a low-foaming surface active agent and lubricant which provides effective dispersing and emulsifying performance for oil, waxes, fats, dyes, and pigments. Due to its unique properties, Turkey Red Oil is used for bath oils and in cosmetics. Other than these, it is also used in the textile industry and sugar industries. • In the agricultural industry, it is used as natural manure and helps plants to grow. • In textiles, as surfactants and wetting agents, in paper industry, in cosmetics, in pharmaceuticals industry • For the soap-making industry that wishes to manufacture transparent soaps.
Another exception is PEG-40 Hydrogenated Castor Oil, but that's no more "natural" than PS80. But if you are interested in trying it, here is a bath bomb recipe that uses it
Other than jojoba oil, most waxes will not work in bath bombs because of their high melt temperature - hotter than any bath you would want to take. Wax also doesn't moisturize; it works as an occlusive to prevent loss of existing moisture.