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.