Although I didn’t find anything on the question about the possible immortality of the animal soul within the compact and well-designed covers of my Catechism of the Catholic Church, Second Edition, I did find a very nice, almost beautiful, entry concerning animals and their purpose in creation:
Animals are God’s creatures. He surrounds them with his providential care. By their mere existence they bless him and give him glory. Thus men owe them kindness. We should recall the gentleness with which saints like St. Francis of Assisi or St. Philip Neri treated animals.
I’ll do a little more research and see if I can find anything more specific.
Can you hunt the prey for the lion,
or satisfy the appetite of the young lions,
when they crouch in their dens,
or lie in wait in their covert?
Who provides for the raven its prey,
when its young ones cry to God,
and wander about for lack of food?
Do you know when the mountain goats give birth?
Do you observe the calving of the deer?
Can you number the months that they fulfill,
and do you know the time when they give birth,
when they crouch to give birth to their offspring,
and are delivered of their young?
Their young ones become strong, they grow up in the open;
they go forth, and do not return to them.