This argument is simple - were we to transplant the conversation into a setting where a florist was refusing service to a black or mixed race couple about to marry, we would intuitively find that harder to defend. The majority of us would find it downright wrong for a baker to refuse a to bake a cake on the basis of his deeply held religious beliefs about miscegenation or the superiority of his own race. So what is different about the case with a same-sex couple? In both cases you have the couple wishing to be free from discrimination on the basis of immutable characteristics. After all, experiencing same-sex attraction is no more alterable than being a person of colour. In both cases you have a service-provider (be that baker or florist) who believes that to provide this service would be immoral. However, in one case we have little doubt that the florist is in the wrong, while in the other we have a higher percentage of people (and indeed a supreme court) willing to side with the baker. Perhaps the reason for this is because more people see same-sex attraction as a choice, something that the lived experiences of millions of LGBT people has time and time again contested. Even if it was a choice, it can be argued that so is marrying someone of a different race yet we do not hesitate in condemning those who discriminate against such couples. Surely it follows that unless we can show a tangible and significant difference in these two cases, we should treat them in the same way.
The first response to this is to say that perhaps it would be legitimate for a service-provider to refuse service to mixed-race couples should their religious beliefs be significantly strong. The logic of this argument isn't necessarily that our position on same-sex couples should be reconsidered - maybe it is our position on religious based discrimination against other groups that should be altered. The second response is that there is something fundamentally different about refusing service on the basis of race and sexuality. Maybe it is that sexuality is mutable. More importantly though, there might well be something different about the degree of religious offence you are causing when you force someone to cater to a same-sex vs mixed-race couple. Most religions today have no strong race taboo - you can be Christian and black, or Muslim and white. There isn't that same acceptance of LGBT individuals in most religions. Therefore, there may be a significant difference in the act of refusing service based on these two characteristics that means it is more acceptable in the case of same-sex couples because the degree of religious offence you cause is greater.
P1. It would be wrong for a service-provider to discriminate on the basis of race, regardless of religious beliefs. P2. Race and sexuality are sufficiently similar in the case of service-based discrimination. C1. Therefore, it would also be wrong to discriminate on the basis of sexuality, regardless of religious beliefs.
Rejecting the premises
P2. Race and sexuality are sufficiently similar in the case of service-based discrimination. You can contest that there are multiple important ways in which they are different - firstly that they are intrinsically different, and secondly that in the context of service-provision they cause different amounts of religious offence.