There's no denying that gender dysphoria is a handicap and disability. We can't deny that those who suffer with this disorder, would rather be normal. I had a neighbor who identified as a transgender and they told me that their life is a living hell, so I know their lifestyle choice comes at a huge cost. It ruins their relationship with their most of their family, friends and relatives and they become a social misfits.
I did try to share the gospel with them, but it wasn't received because as you rightly say they couldn't hide who they are, so they came out and announced it by their lifestyle choice.
Although their choice was not convenient and they knew it would cause them much trouble, they still decided to go with it. We should consider that there are many other sinful lifestyles such as pedophilia which people keep in the closet, because they could be sent to prison for coming out.
I personally believe that God hands some people over to serve their sin;
Romans 1 :28 And even as they did not like to retain God in their knowledge, God gave them over to a debased mind, to do those things which are not fitting;
29 being filled with all unrighteousness, sexual immorality, wickedness, covetousness, maliciousness; full of envy, murder, strife, deceit, evil-mindedness; they are whisperers,
30 backbiters, haters of God, violent, proud, boasters, inventors of evil things, disobedient to parents,
31 undiscerning, untrustworthy, unloving, unforgiving, unmerciful;
32 who, knowing the righteous judgment of God, that those who practice such things are deserving of death, not only do the same but also approve of those who practice them.
I believe this passage of scripture, is saying that the reason people chose sinful lifestyles is because God gives them over to a debased mind. This must mean that He doesn't give His people over to a debased mind, so they are kept safe from these things.
So you are right, they don't freely choose this sinful lifestyle, they choose it because they have been given over to do those things they would rather not do. Sadly my neighbor committed suicide as a sad and lonely person who was dead for around a week before their body was discovered. I was very sad to hear of their death, because they were very likable and pleasant to speak with. I certainly wasn't judgmental towards them, I actually prayed for them but they never found it within themselves to repent.