Can 2 thermistors give in & out temps

My question.  I have used a double pole/double throw switch to send the outside temp sensor, and another temp sensor on the inside of the truck to a Gentex mirror with temperature readout.  It doesn't seem to work as I had hoped.  I am getting an accurate outside temp reading when the switch is in the outside position, but when the inside is selected, it is very slow to show the increase.  (Current outside is 39 and about 70 inside the cab.)  I waited (probably 10 minutes to get a 4 degree difference on the inside reading and then switched to the outside which quickly gave the correct (lower) reading.  Upon selecting the inside, it starts again at where it was reading the outside temp and very slowly starts to climb.

Is it possible to do what I am trying?  I have a basic understanding of the thermister, but don't know why this isn't doing what I want.  Maybe a "cheap" sensor for the inside?  (It is aftermarket, and the outside sensor is OEM coming with the truck.  Maybe the Gentex circuit is where I am having a problem, perhaps sampling over a certain period of time to help dampen the readings, so not to react to a short stop at a stop light which probably increases temps in the sensor area when the vehicle is not moving.

Any thoughts or help would be appreciated.

I know that Nissan implements a filter on temp displays to keep the temperature from drastically jumping around.

There is acutally a set degree amount the display can change per minute.

Underhood temp probes are not exactly perfect.  They have to account for engine heat and wind chill.  If there was no filtering the temp display would bounce around all over the place.  So, they start out with a base temp and use that as a reference.

I'm guessing that this is what your display is doing.  This probably won't work good on an amp as they tend to heat up quickly.