It doesn't matter if your religion hates LGBT people.
Similarly, it doesn't matter if your religion hates Muslims, or people who eat pork, or Catholics, or black people. You don't get to deny service to someone in a head scarf, or wearing a cross. Your faith thinks women should be seen and not heard? You still don't get to deny women service.
If you run a business (which is licensed by the state, remember), you can't choose whom to serve .
And the courts in CO agree. Says the judge,
Compelling a bakery that sells wedding cakes to heterosexual couples to also sell wedding cakes to same-sex couples is incidental to the state’s right to prohibit discrimination on the basis of sexual orientation, and is not the same as forcing a person to pledge allegiance to the government or to display a motto with which they disagree. To say otherwise trivializes the right to free speech....And because baking a cake is not part of religion, or speech,
Respondents’ refusal to provide a cake for Complainants’ same-sex wedding is distinctly the type of conduct that the Supreme Court has repeatedly found subject to legitimate regulation. Such discrimination is against the law; it adversely affects the rights of Complainants to be free from discrimination in the marketplace; and the impact upon Respondents is incidental to the state’s legitimate regulation of commercial activity. Respondents therefore have no valid claim that barring them from discriminating against same-sex customers violates their right to free exercise of religion. Conceptually, Respondents’ refusal to serve a same-sex couple due to religious objection to same-sex weddings is no different from refusing to serve a biracial couple because of religious objection to biracial marriage. However, that argument was struck down long ago in Bob Jones Univ. v. United States.