Is it permissible, Islamically, to spend money on treating a sick cat, and is there a reward for that, knowing that the cost of treatment may be high and entail sacrificing some luxuries?

There is no harm in giving the sick animal the aforementioned injection. Attempting to treat it with this new medicine is an act of kindness towards it, and both of you will receive reward and recompense, for the Prophet, peace and blessings be upon him, said: "In every wet liver there is a reward."