shame?!?!?! Wtf?!?! Utterly bonkers. And as for needing professional help, err, no, she really doesn’t sound anything like a child in need of serious professional help. What an odd life that poster must have led to be under the impression that a five year old child lying occasionally is grounds for professional intervention.
Utter balderdash. Coldswallop. Nonsense.
Anyway, back in reality, OP, I came on to say that this isn’t about you! 
By which, I mean, your dsd is lying to her mum mostly whilst in her mums care right?
So, I do actually think that your dsd is really wanting a reaction from her mum, and whether that’s a little girl trying to say what she thinks her mum wants her to say, or a little girl needing attention from her mum, that’s really something only her mum can divine and respond to.
But that does mean that the punishment does mean very disjointed coming from you guys in her other (main) home.
So it does feel really important that not only do you dial back the severity of the punishments but you take a good look at how to connect any punishment to the deed itself. Any punishment or consequence becomes ineffective if it’s not linked in the child’s mind, and one that doesn’t happen in the same context becomes miserable without benefit for both you and dsd.
Currently there’s no link in terms of:
- place
- people
- time
- topic (doll does not = content of the lie)
- or causality (losing doll does not = real world consequence of a lie)
Especially for a five year old, they just don’t have the mental development for a punishment like this to have any effect, which is exactly what you are experiencing.
Hopefully you aren’t going respond by getting into a cycle of punish harder and harder to try and get the effect you desire, because it’s just not physically possible for your step daughter to display the required change in behaviour. I really think you’ve learned this throughout this thread already, but I’ve seen it happen in other situations and it’s so sad when that happens.
She just cannot learn the lesson in the way you are currently trying to get her to learn it, but, I would shift the emphasis away from punishments and towards addressing the root of the problem with her mother.
Good luck 