i think (and according to the bible)that what is being proposed is healthy in keeping the population of these cats in check, healthy, and in this cases more tolerable.
as stuartist of the earth it is our job to keep populations like this in check. We are to control the numbers of animals (in this case cats) so disease and sickness do not kill them all off (and any indiginous wildlife and any actual pets that so belong to the people), we are to use the animals for our needs (companionship?), we are to control animals in a manner in which they are not harmful to people, and finally we should protect them from over-killing and abuse.
you cant make a stray cat people and family compatible, it just doesnt work. plus the money needed to fund a shelter supplies, medical supplies, food, water, the cost would be rediculous!
Scripture does not indicate a principle that would assume it as a sin or ungodly practice. It does not produce an evilness of man nor does it produce that attitude.
However by some of the comments im reading in this thread i am seeing an evilness produced, namely from you of the fact that you would "kick the crap out of them" an unjustified act which is not a characteristic of a christian.