Alright, let me try and take stock again.
You have four central elements in play:
"I want some reason to behave morally and to check what is moral or not regardless of what the person feels."
"If [and only if] people already have a motivation ... THEN it makes sense to talk about actions that are right and wrong"
"Let's assume the psychopath is religious then he would have access to this whole moral discipline and he would have a reason to follow it regardless of his ability to feel empathy."
"But moral claims only makes sense if [someone intends] to do good for what ever reason. Without a intention to do good it is a meaningless statement."
So 1) you're willing to grant that moral obligations hold if they can be shown to be objective, but 2) you don't think that can be done, and 3) you're willing to count something about religion as such an objective moral reason. Finally, 4) intention plays a critical role in morality.
I don't think you can maintain all of those without problems. The first one that comes to mind is that you outright reject any possibility of (1) by adopting (2) or (4), which defeats the point of insisting that I provide you with (1). Throughout I've suggested that (1) might not be possible in the first place, that that isn't how morality works, etc. I've given you reasons to think so, and you haven't given me reasons not to. In terms of justification, your position that (1) is necessary for morality is lagging behind.
Another issue that springs readily to mind is that you contradict yourself by maintaining (1) and (2) along with (3). You ask for an objective reason, give yourself one (religion), then turn right back around to (2), rejecting your own objective reason. You're simultaneously maintaining that religion is and cannot be (for someone doesn't have a desire to follow it) an objective reason.
While we're on the subject (and you've been doing this throughout, but I haven't addressed it head on yet): What makes you think religion or god is an objective reason to be moral? What's the difference between that and an empathetic understanding of others' pain, or whatever other moral consideration that can be brought to bear on a situation? Especially if (again, as you've been saying throughout) God's commandments only seem to bear on his followers/those who desire to live up to them. When I pressed you on this point earlier you didn't give me a coherent answer. You maintained that actions have consequences, that his commandments affect everybody, but that everyone only has to do what they themselves desire and intention is necessary for morality. When I asked you about the fairness of this, and how you could explain that a person can be punished for doing nothing wrong, you shrugged the question off and dismissed it as a matter for theology. You can't save an inconsistent position by saying "I'm sure there's a way theologians can theorize their way out of this." Surely you see what a lazy response that is. If a position ends up consistently producing paradoxes and contradictions, the way to go forward isn't to say "I'll wait it out - an answer will surely come someday," it's to critically evaluate its foundations and reject them if need be.
Another criticism of the idea that religion/god is an objective reason: What if someone doesn't desire to please god, doesn't desire to believe in him, and absolutely doesn't care if he's punished for eternity? Suppose this person is a masochist who would enjoy being punished by god. Your supposed objective reason wouldn't bear on this person at all. But perhaps you maintain that it would regardless. Yet if what's behind the objective reason isn't desire to please god nor a fear of punishment, then what could it be? Nothing related to desire. Given this case, then, you'd need to either keep the desire-morality relationship and therefore drop (1), or insist that there's something objective about the religion/god reason and drop (2).
I've gotten this far without addressing religion itself, but I feel I have to at this point. Much of how you're thinking about morality is intimately tied up with metaphysical ideas about the existence of god, what god's like, the existence of heaven and hell, etc. I'm afraid we're not on the same page in terms of those things. I'd request that, if you're set on basing your morality on those things, you give me reason to believe such things exist in the first place.
This discussion is long enough, and if you don't feel like moving into that subject, I can't blame you. But I insist you think carefully about what grounds those beliefs about god and the afterlife before you use them to think about morality.