you sound very torn

Kids are very good at finding ways to play parents off each other, so you and dp need a united front. He might need to step back a bit, but it sounds like you need to step forward too. So if dd is tantrum ing, he needs to support you and not have a go at her itms? I think he also needs to spend some time with her one on one. Children respect their carers when they know they can trust them and they are there for them.

I think it's problematic to expect your dd to not interrupt you and dp in another section of the house unless there is another carer for her. Is it possible to get someone to help you out with her each day?

I think her behavior will improve once she feels there I harmony in the house. Ime hostility only breeds hostility. Your dp needs to put his feelings aside and try and accept her for the innocent child she is....it's hard when he feels she is ruling the roost, but she is just trying to find some security.
In the same way Positivity breeds positivity. I wouldn't focus on the lying but rather on all the great things she does and when she's honest really praise her.

Good luck it sounds tough. I hope your dp is ok too. It sounds like this might be a bit much for him too.