If you examine the schematics of those regulators, you'll see that they buffer an internal reference voltage with a darlington. That darlington can source current, but there's really nothing in there that can sink a significant current. It's different from a pair of stacked batteries, in the respect that the batteries can sink current, but the regulators can't. Not even a positive and a negative regulator will work adequately, since they'll not always agree on what they're doing, which can produce sometimes unpredictable and always undesirable results.

Your input voltage is from a battery, and not from a rectified and filtered AC input. As a result, you really don't need the regulator features that mitigate the input ripple and other input-induced effects. In theory, your input supply is essentially "dead quiet." A power op-amp, with sufficient (meaning considerably more than your output requirements) output power, can produce a pseudo-GND that will support a moderately low-current system consisting of two zener diodes, or two transistor-buffered zeners. You could even use an op-amp to buffer each of the two zeners, or even use two op-amps to generate that bipolar output with a single zener reference, one as an inverting amp, and one as a buffer.

I'd suggest you look at datasheets for very old voltage regulators, e.g. LM305 and uA723, not to copy them, but merely to understand the techniques involved.

RE