Right, nobody has a right to a job. They can choose to starve if they want. Is that what you're arguing?
The reason it seems that I'm asking you the same question over and over is because I want you to stop using the passive voice. Don't say "she has to change her business plan." If you really believe what you are arguing in favor of, say "She should be forced to have sex with the black guy even though she doesn't want to."
Look, the mistake you're making is very simple. You are equivocating between private arrangements and lawful authority. When a private employer tells a woman who doesn't have a lot of good options "Have sex with this man you don't want to have sex with, or else lose your job.", that's obviously coercive. She is being coerced to have sex, because of the unequal power relation between the employer and the employee. Duh! Even some crazyass libertarian should be able to see that. But when a government tells a woman who doesn't have a lot of good options "Have sex with this man you don't want to have sex with, or else lose your job.", that's not coercive. In that case she has a free choice about whether to have sex with him -- because it's the government, not a private employer, see? Besides which, the government doesn't have power over her, because if she doesn't like it she can vote them out. And they deserve their power over her because they represent the Will of the People. And even if it is coercive sex, that can't possibly make it rape, because rape is immoral; and what they're doing to her isn't immoral because nobody whose opinion matters feels sorry for racists. Q.E.D. I can't imagine why you're having so much trouble with this concept!
Look, the mistake you are making is very simple. You are missing the distinction between enforcing an action, and enforcing justification for that action.
There is no law that says anyone has to have sex with anyone else. But there is a law that says that a person who has a business selling anything to anyone may not refuse to do business with somebody for one of a handful of defined reasons - and race is one of those reasons.
The law does not coerce prostitutes into having sex with any individual; but it does say that it is illegal for her to decide,
in advance, to refuse to have sex with people solely because those people are black.
The offence is not the refusal, it is the
reason for the refusal. The power relationship is not at issue; the government is indeed imposing its demands on prostitutes - but those demands are not "You must have sex with partners other than those you would choose for yourself"; they are "You must not use racism as your criterion for selecting clients".
Your distaste for governments doing their job of telling citizens where the boundaries of their freedom lie, (and enforcing those boundaries), is a separate discussion; if we accept that governments can and do set some rules, then there is nothing in this anti-discrimination law that endorses rape, any more than laws prohibiting bank robbery endorse poverty. If we do not accept that governments can set any rules, then this entire discussion is moot - In that case, the government can't tell anyone not to have sex with anyone they like, consenting or not.
People can engage in any business they choose; but if they wish to also be racists, the government can and will debar them from bringing their racism into the business environment, and gives them a choice between being racist and being in business.
You can be an overt racist OR a restaurateur. If you try to be both, the government will shut your restaurant.
You can be an overt racist OR a shopkeeper. If you try to be both, the government will shut your shop.
You can be an overt racist OR a bus company. If you try to be both, the government will prohibit you from operating bus services.
You can be an overt racist OR a prostitute. If you try to be both, the government will prohibit you from engaging in prostitution.
Notice that nobody is forced by the government to cook a meal, sell a product, operate public transport, or have sex. But they cannot provide any of those things for sale on a 'whites only' basis.