do we believe her?

Iam64 not necessarily that she lied. The story told by Lucan and his friends was that he disturbed someone in the house who was attacking his wife. She had extensive head injuries, it isn't impossible that at some point she lost consciousness, and other things happened she was unaware of. Of course the story she tells is entirely what she believes happened and she has told it many times. But ask most people about a time when they were injured in an accident and they will admit that they have bits where things are unclear.
He was at rock bottom and disappearing for a bit might have seemed a way out of everything
Apparently he is damned because he tried to get his wife treatment for a mental health problem and damned because he didn't deal with her problems.
Interestingly she has continuing mental health problems, blaming this on the drugs she was prescribed.
She has a website www.ladylucan.co.uk/photo25.htm

The fact she has longstanding mh problems doesn't mean she was unable to identify if her assailant imo. Also
Lord L and his pals would say that, wouldn't they? If he wasn't responsible I struggle to understand why he rN, rather than lawyer up and fight his corner.

Nothing to do with her mental health problems, everything to do with the blows to the head, possible concussion and creating false memories.
Maybe he just felt it was easier to leave. Maybe he was persuaded it would be better.

But if he was so keen on keeping his children why not stay and fight?

He left a letter asking Shand Kydd to look after his children before he fled.

Many people run away from crime scenes, you cannot assume guilt because they did so.