I believe the cat could represent Seichi's emotions, both generally and towards his mother. Weve seen Oshimi use the cat to express Seichi's suffocation and abuse by his mother in early chapters (literally having Sei's head on the cat) but now that Sei was forced to see his mother again for a prolonged period of time he was able to settle his emotions (this being represented by the cat sleeping near his mother) as well as the cat walking out of the door and into the sunrise, I believe this represents Sei's "new start" in life after being able to let go or at least come to peace with his resentment of his mother's actions. I got the impression his "take care" was less expressionless and more so an exasperated farewell to his mother after finally lifting the veil and truly seeing who she is. He was able to gain more understanding of who she was and why she became the way she is, I think the cat and the overall theme of this chapter was more about acceptance that Sei desperatly needed.
Of course this is just my interpretation, I read these as they come out so my memory of the earlier chapters can be a little muddled and i may be misreading the authors intentions. This may also sound a bit pessimistic but i truly hope the story doesnt end in forgiveness, I love the dark, surreal, psychological themes Oshimi has been able to convey in this story and i hope that the ending embodies the same feelings from earlier chapters.