Who advised that that chip would work in the application you plan?
the data sheet is not the easiest to understand that I have worked with...but.....
I see a few problems, as this is a display driver chip, and not an Analogue to digital converter!
The chip is a constant current sink
. The current it sinks is fixed by the ratio of R2 in you diagram, as in 12.5/R2. This is a fixed ratio, so will give a fixed current sink.
Does the temperature gauge work by sensing voltage, or current? (Is it measuring the voltage on the wire to it (the temperature sensor is one leg of a potential divider), or measuring the current flow through the gauge (gauge is an Amp-meter)).
If it is measuring current:-
It does not matter what value of resistor you select for each step, the chip will sink the same current, so the gauge will read the same value!
If temperature gauge measures voltage, then you have a chance..... but:-
You are assuming a fixed source voltage from the gauge. However applying different resistances to ground will change the gauge source voltage.
The value of the resistors needs determining in conjunction with the equivalent internal resistance
of the chip (giving the current sink). Actual resistor value = measured resistor value - equivalent internal resistance
To complicate this more, the temperature gauge voltage will be changing as each individual resistor is switched in (via ohms law) I = constant R = changing, so V=IR Voltage must change with each resistor switched in.
However it is more complicated than that, as the effective internal resistance of the chip is not fixed, and will depend upon the voltage from the temperature gauge (it is a current sink so (ohms law) R=V/I as I is fixed, then as V changes R must also change) The chip is a current sink, not a fixed internal resistor.
As said the data sheet is not clear (gives no internal circuit representation), but a Comparitor used to sink current, as shown, will have constant current characteristics.
Build it and prove me wrong
I think to use this chip you will need to use it as a driver for 10 transistor switches (or JFET much lower ON resistance) or CMOS analogue switches. These will work fine with the constant current characteristics of the 3914, and still switch the resistances you require.