BWT: is it possible to predict the next character in a sequence?
In general, no, but it's difficult to prove a negative given that there might be some data structure equivalent to a BWT that provides the prediction capability.

The L vector in the BWT is simply a list of the input string's characters sorted by their following context, i.e. the suffix that begins immediately after each character.  Its information content is inversely related to how well this following context predicts the character preceding it.  

This predicability is present in many types of text and in many models for generating text, but it's easy to fool adversarially if you have some algorithm like you describe in your second paragraph.  For instance, if your algorithm has only read a portion of the input (you don't specify how it operates, so I'm guessing), I believe it's possible to extend a string in such a way as to spoil any context where you think the prediction is 100%.