I can't remember a time when DH or I didn't bend over backwards to do our best for SD.  But I often hear a twisted version that makes her the victim

Last fall, she needed a knee replacement.  Since she has alienated everyone else in her life, DH and I were the ones who took her to the hospital, stayed during the operation, brought her home and took care of her.  We tag-teamed for a few days with him spending several nights.  I grocery shopped, cooked, cleaned, etc.  But the version I heard from others was that we pretty much left her to fend for herself, didn't provide a cleaning lady as the nurse supposedly advised, weren't concerned.  

This was just one incident of many where our actions are twisted to make her the victim.  I tune it out as much as possible.  I don't believe one word she says and anybody else who spends time with her feels the same.

But it is hurtful to deal with and makes you lose your trust.  Our SD has always been like this.  Is this a new thing with your SD?

I would not offer any further help to someone who treats you like shit and lies about you. 

You don't fix things with a toxic SD by chasing or bending to their will (dysfunction).  She is using a smear campaign to shame her father into doing her will.  If you worked with this person you would distance yourself, danger signs flashing. Same goes for a grown skid.  Do not invite drama into your life by continually responding to it.  " Sorry you feel that way but that's not what happened, when you can treat us with respect we would be happy to see you again " is the only response to SD. Other people (flying monkeys) need to be told the truth in a matter of fact way then move on. Drama triangle in action.  SD baits BD into defending her and puts you on edge,  SD gets out of facing you again and possible discord between you and your daughter.   Basic game and once you see it , you can not play it anymore.