Seeking advice for an alarm system

Hello! I'm currently working on a project regarding hyperthermia. I am to create a wireless alarm system in which one device is to measure the temperature of the interior of a vehicle, as well as measure the distance between the first device (transmitter) and the second (receiver). If the temperature is higher than a certain number (say 85 degrees Fahrenheit) the alarm will go off. If the distance range is above 10 ft, the alarm will also go off. There will be an off button.

I am very interested in using the Arduino nano, as it is small, and I need to make on of the devices small enough to fit in a large key fob. I will also be using a ds1620 for a thermometer.

Is this the right way to approach the project? I know little to nothing of transmitters and receivers, so any advice would be helpful!

Are you planning on leaving your dog in the car for an extended period of time?

The main challenge I see is to find a way to measure the distance. The most obvious techniques involve the mobile part (presumably the thing you call 'transmitter') transmitting some form of radio signal at regular intervals, so that the receiver can raise the alarm if it stop receiving them (implying the transmitter has gone out of range). However, providing an acceptable transmitter battery life within your packaging constraints could be difficult.

That has been my challenge as well. Presumably the thing I call "transmitter"? I'm not sure what the correct term for this would be, sorry.

I have looked into ultrasonic, radio, and even bluetooth.
Besides the distance, what advice would you have?