From what you say, it does have to be highly regulated, but the absolute value is not as important. From your figures, you imply a stability of better than 1 part in 10,000, over 10 degrees C. Do you have a means of actually measuring this, what is the resolution, temperature coefficient and drift on the meter you have. Some battery types have significant temperature coefficients, so unless you can hold the temperature constant(or you can compensate for it), they will not meet your needs. You need to look up the curves of voltage versus temperature for the various battery types. Load is high, so self heating is not an issue. I think an Alkaline cell is around 1mV/4 degrees C, so you would get away with it if the temperature doesn't shift more than that temperature range, you need to check that out though.
There are regulators and references with noise down at the uV level, as Dean said, good references may have temp coefficients