Solved it. Just took some playing around. Enclose the operation in a lisp command, passing the block name after the BEDIT command and you're golden.
(Command "BEDIT" "ExampleBlock")
So in the context of SendStringToExecute it would be something like this:
document.SendStringToExecute("(Command \"._BEDIT\" \"" + blockName + "\"\n)", true, false, false);