I would encourage you to calculate how much of your current expenses are directly related to the cousin.
Things I would include: higher housing costs for the bathroom being an issue, food costs, car costs, utilities costs (electricity and you have mentioned really long showers so water costs too), and any other costs you are providing to him.
I am all for helping out family, but it seems as if you are shelling out $800-1500/month for this relative to live with you, if not more. Sometimes you need to put your own oxygen mask on before you can help others. If having this kind of money every month back in your pocket EVERY MONTH would allow you to spend time with your family or get closer to your financial goals it could be worth considering having different conversations with your husband and his cousin. He is not just couch surfing and using minimal resources, and based on your shower comment and your comment around him helping with child care and cooking I expect he is disruptive to your family and unwilling to help out around the house for the privileges he is receiving. He should be doing a hell of a lot of work around the house for that kind of money. If you are paying someone $15/hour this would be equivalent to 53-100 hours/month.
If there is a light at the end of the tunnel for you with this family member, maybe a few more months is worth it. From my perspective it seems as if he has no motivation to move out and be successful on his own. You moved houses to accommodate him, this is not a normal thing to do for temporary assistance. You are paying him through the business and not allowing him to understand his financial responsibilities of being an adult. If I was in his shoes I would in no way be considering going out on my own, it is way too comfy of a position to live with you.
I am making some assumptions in my post, I am sure there are some other reasons he is living with you that you have not posted here. But from my perspective that is how things look and it seems to be causing you a great amount of stress. Put the facts in front of your husband so you can talk about them. His cousin needs to understand what a privilege it is to live under your roof and start learning skills to be successful on his own.
I do not mean any of this in judgement, my mom's friend just went through what seems like a similar situation with her son. The son was home using all of his money to fund hobbies and not become a real adult. He would eat all of the food and never replace it, take extremely long showers, leave messes all around the house, go to work and spend all of his money on computers and guitars. His mom didn't want him to fail on his own (he was only working part time, not enough to cover all of life's expenses), and his dad wanted him out so he can learn the hard way that you need to take responsibility for your life and rent isn't free. Them kicking him out on his own was the best thing for the son, he found full time employment and sold some of his electronics and guitars to pay for food. It reduced the stress on my mom's friend and her husband because they were not constantly arguing around what to do about their son, and they had the freedom to live in their house with the rules they choose.
I think it is totally fine to provide assistance to family, just understand what it is costing you and ensure you are okay with the costs.
A crazy thought: another option could be to have him move out and still provide financial assistance for a period of time? Maybe $200-500/month for the first 6 months as he gets out on his own? This might be cheaper for you, he can still work for the business, and he can start to learn how to live on his own. I would try to look at other options that could work better for your family.