Whether mentioned in Scripture or not, the transgender movement clashes
with traditional Christian theology that teaches the only God-given
expression of human sexuality is between a man and woman who are
married. (Chrsitianity Today, The Transgender Moment)
There's a lot to unpack here so let's take it one at a time. To be clear what we are talking about, let's mention that when this objection is brought up, the speaker is thinking of homosexuality to be attraction to a person with the same sex as yourself. More on that later. In this article I'll use the example of a transman for simplicity's sake. Of course a transwoman works the same way with every sex/gender reversed.
First, transexuals can have any attraction: to men, women, both, neither and possibilities go even further. An asexual transman is certainly not homosexual so presenting transexuality and heterosexuality as endlessly at odds is mis-informed as in Christianity Today. And many transexuals regardless of their orientation are not sexually active which makes them chaste regardless.
Now, suppose the man is attracted to females. The speaker believes this is homosexual because both people are (originally) female. The transman believes this is heterosexual because he is a man and she is a woman. (Notice the changing use of "female" and "woman.") In the article Objection: Sex Cannot Change, I consider whether the transman should be considered a male or female. This objection comes down only to what we think about sex. Therefore, it may be more productive to talk about that topic instead because it removes one layer of complexity.
There is another way to deal with this issue. Many people and myself included do not believe with the speakers idea about sexual orientation. We replace sex with gender in this idea so that a homosexual is a person attracted to a person of the same gender. Therefore, the transman is heterosexual because he is a man and she is a woman, regardless of whether he is male or female. This will satisfy a few people that are willing to believe sex and gender are different. Most, however, don't believe in differentiating sex and gender at all so they won't do it for sexual orientation.
There is one final subject. Regardless of what is and is not homosexual in practice, many Christians believe that homosexuality and all transgenderism have a common origin, making them different manifestations of the same root problem. Dr. Milazzo addresses this concept in some depth and I do too.