Why does my $LD_LIBRARY_PATH get unset when using screen with bash?

Because screen is setuid, it unsets LD_LIBRARY_PATH. When your shell is tcsh, tcsh initialisation (from .tcshrc I suppose, since screen doesn't default to creating login shells) sets LD_LIBRARY_PATH again. If you want LD_LIBRARY_PATH to be set in your bash screen windows, set it from .bashrc.