If a government regulation created the contraception mandate, a government regulation can completely repeal it, and eliminate the need for a businessperson to declare a moral objection.
Why Trump isn't going that route, I don't know.
I would argue the First Amendment prevents the government from questioning Americans' religious beliefs. And thank God for that.
But, since courts are ruling in favor of broadening the (probably too restrictive) mandate and since Trump's proposed rule change means that any businessperson could claim a religious or moral objection, the government would have to accept it. The government can't, or at least it shouldn't, question an American's religious beliefs.
Other Americans can freely scoff at a businessperson's claim of religion, but the government shouldn't.
So, rather than have every business owner who feels women don't deserve contraception coverage in the owner's health insurance plan dream up a religious objection, just drop the mandate entirely.