Economy (noun): the intentional saving of money or, less commonly, the saving of time, energy, words, etc. (in this case: rats)
"For economy's sake" here means to be sparing or to be careful with the supply of rats, as there is only one.
If the Prince would actually kill the rat, he would no longer be able to follow the advice to condemn a rat to death from time to time.
I hope this helps.