To expand on Mr. Wayne's skilled assessment, a 4-string bass generally doesn't need carbon fiber rods. This assumes that you are using a good neck material. Rock maple is common not because it is cheap, but because it is a good neck material. There are other good materials as well, of course. Also, if you laminate your neck, meaning you make it out of multiple pieces of wood (glued together with opposing grain patterns), you will add a great deal of rigidity, even if you use multiple pieces of the same kind of wood.
BEAD stringing actually generally results in less tension on the neck than standard EADG, not more. That is, of course, unless you are using something like a balanced string set from Circle K. Their balanced sets are designed to give even tension across all strings in the set. If you aren't using a balanced set, tension on the B string is lower than on the E string. Thus, a BEAD tuned 4-string has less overall tension than an EADG tuned 4-string. In either case, you shouldn't need any carbon rods. They won't necessarily hurt anything, though.
