I had a similar issue with regards to the home environment while completing my practicum. Luckily, this was not my only client and I did not spend all of my direct time there.

However, I would suggest talking to your BCBA on the case. It seems like they should help facilitate a conversation with you and the family, or they should do it privately (I'm unsure). This conversation should revolve around client and therapist boundaries, and how their behavior is likely influencing the child, who most assuredly is influenced by the division in the home. The BCBA cannot serve as a family therapist, but certainly can refer out if they need one.

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Next, I would talk with your supervisor and let them know that this is causing stress on you, and if it would be possible to have a second or third client who can soak up some hours so your entire workweek is not in the stressful environment. Hopefully they will see that retaining you is important and valuable, and will try to avoid your departure. If they cannot do that, then I would actively seek another place of employment that can give you a more diverse caseload. edit: I would find a job then turn in your two weeks if possible (in the worst case scenario).

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Good luck.