All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If you are harmed by the presence of the cat in the house, then there is nothing wrong with you asking your husband to take it out of the house. He does not have the right to force you to keep it; obedience is only obligatory according to what is reasonable and it is not reasonable for a husband to ask his wife for something harmful to her. Imam Ahmad, may Allaah have meryc upon him, narrated in his Musnad on the authority of Ibn Abbas who said: “The Messenger of Allah said: “There should be no harm nor reciprocal harm.”
In addition, if by what you mentioned, you refer to the disease of this cat whether this prevents taking it out of the house because it may be exposed to death, then the answer is that this is not a legitimate prohibition to get the cat out, but the prohibition is to imprison it until it perishes without taking care of it, as the Prophet said: “A woman was punished and entered Hellfire because she kept a cat in captivity until it died; she neither fed it nor gave it water to drink; nor did she set it free so that it could eat from the vermin of the earth.” [Al-Bukhari and Muslim]
In conclusion, we advise you to make discussion a basis for resolving any problems that may arise in your marital life, and that the matter of the cat should not be the subject of a dispute between you, as the matter of the cat is easy and should not be a reason for dispute between you.
Allah knows best.