You're right that there have been few playback studies assessing the response of sperm whales to sperm whale clicks. A couple of studies have shown a response (reduced foraging, returning to surface[1,2]) when killer whale vocalizations are played in the presence of sperm whales.
As an acoustician focused on odontocete echolocation clicks, I field a lot of questions about communicating with whales and when we're going to "get there" or "figure out what they're saying". Based on the title of your question, I'm not sure whether you envision the end goal of playback experiments to be getting us to the level where we could communicate specific messages to the whales themselves. If that is what you mean though, I think we're probably a long way off from that, and I'm not sure whether or not it's possible. It is not a guarantee that the way we use language and the way sperm whales using vocalizations are compatible enough for us to truly communicate.
If you mean communicate more in the context of understanding what types of clicks mean by playing them at the whales and observing their response, i.e., when we play clicks of type A, the animals always come to the surface, so clicks of type A must be some sort of surfacing cue, that's also very complex. 350
For one thing, to draw conclusions you'd likely have to conduct experiments in a wide variety of locations/different groupings of animals, which gets expensive fast. Additionally, as animals are difficult to observe at sea in most cases, you have to factor in the added expense of equipment such as satellite tags that would be required to understand the animal's response to the sound, as you likely couldn't rely on simple visual observation. I'm not saying these efforts wouldn't be feasible, but to my knowledge these are some of the reasons why they may not be more common.
References:
[1] <PERSON>, <PERSON>, et al. "Responses of male sperm whales (Physeter macrocephalus) to killer whale sounds: implications for anti-predator strategies." Scientific Reports 3.1 (2013): 1-7.
[2] <PERSON>, <PERSON>, et al. "Sperm whales reduce foraging effort during exposure to 1–2 kH z sonar and killer whale sounds." Ecological Applications 26.1 (2016): 77-93. 524