This is a tough one. If the church believes in infant baptism, then they could offer to baptise the child privately, but not as a public display in front of the congregation.
During the last century, one of my relatives had parents in a religiously mixed marriage. The parent who claimed religious “ownership” of the child never went through with taking the child to church. So the other parent just started taking the child to the other denomination, and when it was time to be confirmed, the child and her faithful parent had to skulk into the church office to have the 12-year-old baptized. The church was not fooling around, and did not want to encourage neglectful parenting, since they believed that baptism brought God’s protection to the infant.
These requests for special accomodations by gay couples are often a direct attempt to denounce the church, and by extension, the Word of God, and lay the groundwork for lawsuits. So the churches need to address these matters in their by-laws in advance of such requests being made, to remove a legal ground for discrimination against individuals (as distinct from a long-standing, clearly stated policy).
Sadly, you are right. In today's litigious society - especially the Left, the LGBTXYZ people are actively seeking out churches to sue and make examples of thereby hoping to bend Christians to their will or put them out of business.
The early Christians, I don't think, even had to consider actions like this because it was common understanding that extra-marital sex was unholy behavior and sin. Having a clearly spelled out statement of faith and expected behaviors of members give guidelines and standards to point to so there should not be grounds for lawsuits.