The default speed of the LTC2400 is about 6 readings per second. So at 1 µV/ sqrt(Hz), a 6 Hz Bandwidth should give about 2.5 µV RMS noise. So my guess is the measured 1 µV/ sqrt(Hz) value includes some extra noise sources, like from the reference (not measuring at 0) or not so good anti aliasing filtering. It is also possible that noise gets higher if not measuring at 0, due the adjustment procedure. The 1.5 µV_rms specs should correspond to about 0.6 µV/Sqrt(Hz). The conversion from RMS to peak - peak values depends on the bandwidth / frequency ratio. As the readings are essentially independent it's usually a factor of 5 to 7.
There are other integrated sigma delta ADCs with a lower noise value, but often higher INL. Some if the high resolution (e.g. 20-24 Bit) SAR ADCs from LT could be a option, though possibly with 1/f noise. Single reading noise is high, but with some averaging / integration a line period really low noise levels are possible.
Using a dual slope ADC with residual ADC is tricky, as dielectric absorption will severely limit INL, unless an exotic integration cap is used (e.g. Teflon, vacuum). So this would usually need a kind of multi slope variation, e.g. similar to the 34401. This would be more like 20 Bits from the up / down slops and maybe 12 Bits from the residual and some loss due to overlap. It's not just using many slopes, but low loss caps are much easier at 1 nF than at 100 nF.