Yes, there is a strong overlap between anti LGBT sentiments and religiosity but also with conservatism in general. Conservatives tend to put a lot of focus on how uncontrolled sexuality could lead to behavior that is harmful to others and to one self, and tend to be very weary of any expression of sexuality that involves multiple partners, that is casual in nature, that is overly public or is somehow associated with minors.
That said here are the most common conservative non religious arguments against LGBT rights that you might find in rightwing circles:
There is no such thing as LGBT rights
They argue that rights should not be associated with arbitrary political labels and movements. LGBT individuals should have the same rights as any other invididual. By fighting for rights for specific groups instead of individual rights in general, you are likely just trying to justify special privilages and special treatment.
Marriage is about children
They argue that the main goal of marriage is to ensure the child grows up with a mother and a father, as they consider this form of family arrangement to be optimal for child raising as both the mother and father fulfil important roles.
Since homosexual relationships are baren by definition, they see marriage as incoherent as no child will come. When talking about adoption, they will point out that the lack of a father or a mother is suboptimal, as an important role will be missing, and therefore, heterosexual couples should always be given prefference.
They also tend to associate the LGBT movement with casual sex, multiple partners and overly public displays, and they see that as a potential source of danger for the child. They also tend to believe that homosexuality is a learned behavior, and that the child might mimic the adptive parents, which they consider to be an overal dangerous behavior to mimic.
But they tend to approve the marriage of heterosexual beren couples, even if no children will be raised, since they consider it a good practice to emulate marriage behavior. They don't think marriage behavior can be coherently emulated by homosexual couples due to the lack of the 2 roles of the marriage.
Depopulation concerns
One common concern in rightwing circles is the notion that the native population is not having enough children and that this will lead to a cultural death in the future. They generally encourage everyone to have a lot of children as a way to preserve their traditions and values.
Since homosexuality is inheritely beren, this is something that they see as suboptimal, and a potential danger to the future of the culture. This is specially the case since they think sexuality is learned behavior, so, they have concerns that young people might adopt this behavior.
Early association between homosexuality and infidelity
One of the most common scenarios early on that were used against the LGBT movement is the concern that husbands were cheating on their wives with other men, and this lead to a lot of opposition to the establishment of gay bars, and other places where LGBT behavior was normalized.
This narrative lost momentum over the years, however.
Guilt by association
One common narrative is that sexual predators are attracted to LGBT environments, and that they use LGBT labels as a shield. The most common concern is that pedophiles might be hiding within LGBT groups as a way to more easily have access to children, for example through adoption.
They also associate the LGBT label with leftist ideology and communism, so they see the movement as a potential vector of propaganda.
This results in a sort of passive discrimination, where they don't openly antagonize them, but since they see LGBT as associated with a bunch of negative stuff and overall as a red flag so they try disassosiating themvelves from it, for example by not hiring them.