Figure 6 shows 1.2 volts.

True.
But that figure is with the Ref pin connected to the cathode.
When the Ref pin voltage is above 1.25V, the cathode voltage will go lower (but not shown on the data sheet).

The LTC6703 has a supply current of around 10 uA. More than 10x less than the TLV431.

Okay.
But that's only important if you are monitoring really small batteries, like coin cells.
For anything AAA size or larger 100μA is generally negligible.

 

An alternative scheme:

 

You can use Bordodynov's design with a MOSFET also.
Having the LED in the source will increase the turn on voltage, and likely avoid any possible leakage problem when the MOSFET is off.

 

Reduce 47kOhm or use a bipolar transistor with a high current gain. Haven't you noticed that the threshold has changed?

 

Alternative