Is there such a thing as rubber band resistor?

There ABSOLUTELY are such devices, made of conductive rubber of some sort. For example, HERE, from RobotShop.com

I suspect there is a bunch of hysteresis, and that the resistive properties will change over time and use. Perhaps one of the other suggestions would work better

I'm not aware of any "rubber resistors", but there is a thing called a string potentiometer that implements pretty much the same function.

A spring converts a displacement into a force. You can then use a "load cell" to measure that force. Include words like "miniature", "micro" or "SMD" in your search. There are several manufacturers of devices that fit your space requirements.

You could also use an actual rubber band, but a metallic spring will give more consistent and linear results.