The Establishment Clause exists to guarantee the freedoms set forth in the Free Exercise Clause. This is an enabling exception to effectuate the Free Exercise clause as it was widely interpreted pre Employment Division v. Smith. Even this Court, with Kennedy (who wrote Obergefell) would not strike this down on Establishment Grounds. In terms of LGBT discrimination, its one thing to hold that states are PERMITTED to require adoption agencies to work for LGBT couples, but it is a whole different ball of wax to claim that they CANNOT exempt religious people.

This exemption doesn't harm LGBT people in any way. They can still adopt. It simply allows religious groups to operate licensed adoption agencies without also being forced to enable adoptions to which they have sincere religious objections. LGBT people are not losing any rights of any kind.