Probably should have been posted in the prop thread, but here's a suggestion to get you going. I like your idea! And might use it since I have one of those 6-1/2 foot inflatable cats with moving jaw that would be perfect to go along with it. So here's how I would approach it.
Make a tail out of a pool noodle, some stiff wire run thru the middle hole of the noodle and attached to a motor...like one used for an animated reindeer that will travel back and forth. You'll need to figure out a way to "skin" the structure to cover the wire connection points that will probably be visible.
I'd "shave" the noodle down to form a tail. Dollar Tree has them for a dollar and if you want a fatter tail, Walmart carries a monster pool noodle which has a larger diameter. Due to weight concerns the thinner might be the best choice. You should then be able to cut the pool noodle into sections to get some flexibilty in it. Now this is just off the top of my head but maybe some wire like the old wire coat hanger wire would work. Cut to insert in the pool noodle and give yourself enough extra wire to form hooks or eyehooks on each end. The hook area is to be used to attach the next section to keep them together and the one at the fat end of the tail (nearest rat body) would be connected with linkage to the motor. My thought is that the motor would move the fattest part of the tail slowly back and forth and the hooked linkage would move the rest of the tail back and forth as well. You might need to run something like a nylon thread/thin rope from end of tail to the part by the body through the inside of the noodle (thru the eye hooks) to have the tail move as one flexible piece. You need to keep things as light as possible or get a stronger motor.
Depending on how you shape the pool noodle joint area, you might be able to get away without anything to cover the joints. Otherwise I'm sure there's paint you could spray on the noodle or slip fabric over the length to make it look rat like.
I built a large Octopus arm that came out of a hole in the wall of the wine cellar here. Looking into the hole they would just see a short dug out hole going no where, actually this was a vent chimney that did go up to the surface.
When operating the arm would begin it's descent downward inside the chimney, get to the bottom,turn go straight out horizontally from the hole, then turn again upward and even sometimes "Slap" the manacle I had hanging there!
It worked great for 3 1/2 years until a helper allowed a 12 yr. old to wrestle it.
Peoria PBS caught it working on film many years ago as it scared a local 16 year old girl.
The method of construction used steel hinges and cable strung from front to back just as "Wookie" has shown previously here.