Since this issue has been long term and she is getting to the point where she should be outgrowing the inability to deal with her emotions more productively.. YES.. I think therapy is in order.  I'm surprised that she isn't having issues at school...and that they aren't asking for her to be seen.

She's learned to manipulate the adults in her life and keeps it going because it works. You need to hold strong and put her in a safe place to tantrum then walk away and do not negotiate with the little terrorizing demon spawn.  If this is not stopped now you will send into the world  another spoiled rotten female that won't be able to function on her own. Please don't do that.  There are enough out there now.  

She needs to see someone.  She getting older and its getting worst,  this is no way to live. Walking on eggshells in your own home.  Unless you or DH actually do something, nothing is going to change.  Unless you do something different, nothing different is going to happen 

If she has model behavior at school, then she is able to control her emotions and her behavior.  Perhaps she needs more significant consequences and consistency - every time - when she has a tantrum at your place.  Immediately banish her to her room or another quiet place if she doesn't stop when asked and use her currency to come up with punishments that really matter to her and are likely to discourage the behavior.  Make her life unpleasant when she acts this way.  If that doesn't work, then I'd say she needs therapy to learn to cope with not getting her every desire.  

BTW, does she pitch these tantrums at BM's house?  If not, do you know what BM might be doing differently?  

When DH and I got together I bought a program called the Total Transformation.  It's still available I see.  This was 15 years ago and our two groups of kids were coming from different parenting styles.  We needed something to use to help us sort things out.  I always advise from a parent led attitude and as you can see, when the child has control of the home, chaos rules.  Take back your adult power and run your house the way an adult would.  Kids don't know how to handle that power for good of all, only for selfish needs.  If DH is bailing on discipline, there's your problem. 

Try the shock effect. The child has probably never been told what to do and probably won't know how to react when it happens. So, use your "teacher" voice (loud and firm - no yelling) and say "Stop that NOW and go to your room IMMEDIATELY!" Chances are she'll be so taken aback she'll do it. Follow he and tell her she will not be coming oiut of there until you tell her or there will be no TV/Screen time/dessert/phone/whatever she prizes tonight.

Do not try to reason with her because she couldn't care less about what you have to say.

Good luck.

Therapy could still be helpful, but find one that will help with parenting strategies, not just meet alone with SD - you want a family therapist, and have BM go too, assuming this happens at her house. 

She's being rewarded for her behavior with your/BD's attention. How do you guys respond when she melts down? Do you give her attention? Try to calm her down? 

It's an old book but a good one that helped me. 'How to Behave so Your Children Will, Too' by Sal Severe lays out some pretty  sensible and realistic information about parenting thru tantrums. Consistency is your friend.