I did something similar a while before, kind of forgetting the detail, but I think I can give you some pointers on that.
Of course, like creichen said in his answer, it's impossible to get the exact value associate with the LocalVariable, but it's possible to sort of get the whole Abstract Syntax Tree of the bytecode and get know what kind possible value is going to assign to that LocalVariable at some point of the program.
For each LocalVariableNode is assigned with an index, so in bytecode, the value is not statically associate with the LocalVariable because Java allow mutable on that. So in order to know the value associate with the LocalVariable at some point of the program, you have to basically simulate the stack execution of the bytecode (JVM) and then have a table to trace the value assignment for each LocalVariable. The stack execution is through simulate the instruction (Opcode), basically the bytecode. And then you gradually build up a Tree structure to store the Abstract Syntax Tree.
I do have a code for that, but pretty ugly, you can take a look: https://github.com/davidlau325/BytecodeASTGenerator