]I think he does genuinely care for her (as much as he can - I wouldn't like to bet on her safety if it came to a decision between her welfare and his!)

No, and there is the perfect example right in the play - Gertrude unknowlingly goes to drink of the poisoned wine and Claudius says, "Gertrude. Do not drink." He could have stopped her, but knows that he will give himself away if he does so, and so makes the decision to let her die. I think he loves her, as much as he is capable of it, and that having her was one of the motives for the murder, but not enough.