Is expiation obligatory for one who unintentionally caused harm to a bird, then its incorrect treatment led to the worsening of its condition, necessitating its slaughter?

There is no blame upon you for what you did with the cat, as long as you did not intend to harm it, but rather intended to treat it. There is no sin upon you, nor expiation. You will be rewarded for feeding it and showing mercy to it, for the Prophet, peace and blessings be upon him, said: "In every wet liver there is a reward." If you wish to slaughter it for your benefit, there is no harm in that.