I think when David sees Bathsheba he is astounded, breathless, awed, by her beauty, and just her. He is taken by her, by storm. He is in a passion, so deep, he did not know it was within him.
Israel may well have been under orders to kill Hittites, but any foreigner, even one under threat of death, if they abide by the rules of the Lord God of Israel, and even more, defend Israel, are exempt under Mosaic Law. Judaism after the disaster of the Samaritans decided never to forcibly convert people, Yes David was much earlier, but the principle still held, if someone converts on their own, such as Ruth, Davids great, or great-great grandmother, they are welcome. Uriah, being allowed to live, was either a convert, or one who accepted the laws of the God of Israel, and so was allowed to live.
Nathan emphasized the murder, since adulterers are still alive, though condemned to death. The King and his Queen seemed to be exempt, the top just don’t play by the same rules as everyone else. Murder is usually considered the worst offence, as it deprives someone of life, which only God can give, and none should take away, before their appointed time.
Perhaps we just have a different idea of loving someone, and what is forthright. To me, David seems perfectly so, he goes after Bathsheba, like a man possessed. Letting nothing stand in his way, not even the laws of his God. He accepts the rebuke, whether by a palace coup or David’s desires for her child, Solomon, he does become King.