I have given a link to the photography case which sliced and diced the First Amendment.
Simply put you place this case under the Equal Protection clause. I sincerely believe both the 1st and 14th Amendments go hand in hand with one another.
But at any rate, this case I believe debunks the NM court's reasoning on the matter:
“religious beliefs need not be acceptable, logical, consistent or comprehensible to others in order to merit First Amendment protection”
Quoted by Justice Kennedy from the opinion by Justice Burger in Thomas v. Review Board of the Indiana Employment Security Division 450 U.S. 707 (1981),
in the case of Church of the Lukumi Babalu Aye, Inc. v. Hialeah, 508 U.S. 520 (1993)
What I see here is similar to the Obamacare mandating the Catholic Church to pay for abortion. Both of these instances are burdening the religious convictions of faith based groups.
In the case of Jose Merced, President Templo Yoruba Omo Orisha Texas, Inc., v. City of Euless (5th Circuit). The court ruled that the free exercise of religion was "meritorious and prevailing" and that Merced was entitled under the Texas Religious Freedom and Restoration Act (TRFRA) to an injunction preventing the city of Euless, Texas from enforcing its ordinances that burdened his religious practices relating to the use of animals.
So if that is the precedent, the Oregon Public Accommodation Law would be unfairly burdening the Christian couple as they would be obligated to do something that was against the teachings of their faith, they would be forced not to adhere to the doctrine their faith teaches about homosexuality.