The primary issue that comes to mind is the height of most bee balms; they typically get to be between 2 and 5 feet, which would be quite tall for a hanging basket. However, there are dwarf varieties that max out at 16-18 inches and could work in a container.
As for the matter of whether they’ll overwinter in the hanging basket, it largely depends on your climate. In areas with frigid winters, you’d need to protect the basket from winter winds and negative temperatures by insulating the basket. Bee balm also prefers moist, well-draining soil, so you’d need to make sure that the soil doesn’t dry out during summer or get too wet during winter.
Overall, it may be worth attempting with a dwarf variety, but we wouldn’t recommend putting all your bee balm in one basket, so to speak!