And I dispute that those who choose to ignore Biblical condemnation of homosexuality, either because it refers to them personally or just because they think there's nothing wrong with it, are somehow less true believers than those who choose not to cherrypick it out of their belief system.
Then your belief is contradicted by the empirical evidence. There are numerous studies on the subject. They measure people's actual certainty of belief and value and importance they place on those beliefs and on acting upon them in their daily lives. Those are qualities of true believers, because when you believe it, you actually act on it and do so with confidence. Such measures are typically referred to as "religiosity". The fundies who take all of the Bible seriously consistently score higher in this religiosity. Cherry-pickers tend to be filled with doubt and uncertainty even about the most basic foundations of their religion, such as God's existence, and they report their religion as being less important to their daily life and decisions. IOW, they are less religious and less Christian in nearly every sense it which it actually means something to utter those words.
You can be against slavery and still call yourself a true Christian.
You can be against slavery and still call yourself a true unicorn. That doesn't make you one. Whether people utter the sounds "I am a true Christian" has no relevance. The question is what is the actual state of their beliefs, certainty in those beliefs, and importance they place in using those beliefs to determine their identity and choices in actual life. There is rather clear scientific evidence that these cherry-pickers, and especially politically liberal cherry-pickers are barely even theists at all and rarely can be accurately categorized as "Christian" due to their own admitted lack of importance they place on any ideas or actions that are at all particular to any form of monotheism let alone Christianity. Of course their are always exceptions, but as a reliable general rule people who say "The Bible doesn't condemn gays", rarely think about religion or God and the implications for their lives, and are much less confident in whatever cherry picked beliefs they do have.
While I agree with you that it would be great if they'd just dump the religion entirely, the LGBT Christians don't have that motivation. It's not what they're going for, so doing that would move them away from their desired goal instead of towards it. They like being Christians and they want their Christian faith to embrace them. They're not looking to abandon it over disagreement about one point.
IT is not just one point. It coheres with the general authoritarian ethics and intolerance that is the foundation of the monotheistic creator God concept. The people that want to ignore the specific issue of homosexuality and the Bible, almost always want to ignore most other aspects that show God to be an intolerant bigot with self-serving rules, which means 95% of the Bible. That is why such people report that they hardly ever read or think about the Bible, or even think about God and what God expects of them. IOW, they don't think Christian, they just utter the phrase "I am Christian" when asked.
Well, how many passages of the Bible do you need to ignore to make the claim that God is against slavery? How many passages do you need to ignore in order to say that one can be a Christian and believe in equal rights for women? Compare that to how many you need to ignore to say that God doesn't have a problem with homosexuality. I'm no Biblical scholar, but a quick google search shows the number to be around six and the first ones are themes which get brought up time and time again. Unless you're making the claim that none of the religious leaders of the anti-slavery movements were true believers in Christianity and that there are no fundamentalist women who feel that they are the equal of a man, I really don't get why it is that you feel that a topic about which the Bible is essentially uninterested in and only brings up a few times in passing, mainly while talking about other things, would need to be a sticking point of one's faith. You don't need to ignore 95% of the Bible to ignore what it says about homosexuality. You need to ignore parts of six passages out of over thirty thousand, so that's 0.0002% of the Bible. There aren't very many things which the Bible mentions that you need to ignore less in order to cherry pick it out of your belief system.
You can read hundreds of articles by Christians about how homosexuality is completely compatible with their faith and how acceptance of it is an expression of God's love - yada, yada, yada. All of those people are real Christians and true believers in their faith. One does not need to have a fundamentalist label in order to have a Real Christian label and, even if one did, homosexuality isn't a topic which the Bible particularly cares about so it's not one that Real Christians particularly need to care about either.