How might the history of the South and of the entire United States have been transformed had the filibustering efforts of men like William Walker been more successful?

First, if William Walker and other filibusterers had been successful, it is very likely they would have received the support of the federal government (indeed, Franklin Pierce recognized his conquest of Nicaragua as legitimate). Because Walker intended his conquests to be slave societies, this would have occasioned even more debate about the spread of slavery than was already taking place. Some people at the time believed that people in the Upper and Middle South would sell their slaves into the region, thus "draining" the upper regions of the South of slaves.
It could also be claimed that having territories in Central America might have dampened Southern commitment to securing protection for the institution in the North American West. It is difficult to know if this would have happened. What we can say for certain is that the support of many Democrats for these operations demonstrates the commitment to many in the South to expanding (not just preserving) slavery. So I would argue that Walker's success would have just contributed even more to sectional tensions between the slave South and the free North, tensions which ultimately resulted in secession and civil war.
https://www.historynet.com/william-walker-king-of-the-19th-century-filibusters.htm