Do they have a CO stipulating they exchange every other day? If not, I would do exchanges every week or even every two weeks. It will be more consistent for your SD and will limit the interaction BM has with your SO. If she starts blowing up his phone, limit her contact to emails. She harasses him at pick up and drop off, they start doing exchanges at the police station. There are loads of boundaries that can be placed on her that limits her influence in your SO's time with his daughter.

Also, you need to accept that this woman doesn't want a relationship with you, and your SO doesn't have to co-parent with her. He can parallel parent, and that will be easier with an every week/every two week schedule. She isn't immature for wanting to meet you; it's just her preference. Calling you names or trying to convince her DD to hate you is the immature part.

You have to realize that high conflict exes have an entirely different set of rules when parenting. There is no working together or hashing it out. There are boundaries and strictly-written COs. There is no civil and polite; there is avoidance at all costs. The conflict created in these situations is so intense that the only solution is to minimize contact to minimize conflict. You cannot compare your relationship with your ex to the one with his ex even if she were behaving more agreeably. This is especially true since she is high conflict.

...with everything lieutenant_dad said. If I were you, I'd limit all contact with this woman. I would also encourage your SO to limit all contact with this woman. Where possible, they should do pick-ups / drop-offs at a neutral location, or even arrange to pick-up / drop-off when the other parent isn't there - like at school. I would also not worry what the woman is saying to the child about you. You can't control what she says and it sounds like she's not above lying. Just focus on your relationship with the child. Be welcoming and kind, but don't take offense if the child acts hesitant around you. 

This woman will never be your friend and your SO does not owe her a personal relationship - no matter what she tells him. Your SO should focus on his relationship with his child only and cut his ex out as much as possible. My DH is so much happier and so much calmer now that he ignores BM 95% of the time. When I met him she was calling him daily to either berate him or complain about her life. He thought that it would make it easier on his children if he let her mistreat him. She was already remarried at the time and I put my foot down with my DH telling him that I would not make space in our relationship for him to serve as an emotional outlet for another woman. I also cut her off entirely after she tried to spread lies about me with DH's family. 

One word of warning, it will be impossible for your SO's child to not be impacted by her mother's crazy behavior. It will be difficult at times - for you as a stepmother - to deal with that, so don't totally discount that fact. Dealing with a crazy ex wife is no picnic. Her craziness will seep into your life whether you want it to, or not. 

Read the book Say Good-bye to Crazy, it literally saved my sanity.

and destroyed. It is not immature of BM to not want to meet you and while I understand your good intentions of the blissful SM/BM friendship and team for the good of the Skid ..... never forget that the Brady Bunch was a sitcom and the movie Stepmother was a Hollywood script.  The dream outcome in blended family situations is rare.

I completely concur with the suggestion that DH needs to get a revised CO with an EOW or EO2W custody split.  This gives the Skid the best stability considering she has a toxic and high conflict BM.  It also gives  you and your DH time for  yourselves and  your marriage while both your daughter and his are with their other parent.  All Skid and all Blended Family all of the time frequently is the death knell of a blended family marriage.

Though it took us a number of years I was finally able to get my bride to embrace the time that she and I had when SS was on SpermLand visitation. For years she refused to vacation or do much of anything while SS was in SpermLand because she felt guilty that he might miss something.  My bride was the CP and the SpermClan had 7wks of long distance visitation per year. 5wks summer, 1wk winter, 1wk spring.  This gave them dedicated time with the kid, it gave us 3 times per year where we could just focus on us, and it gave us a set schedule and sequence of events to work through when dealing with the SKid's pre-visitation behavioral nose dive and his post visitation behavioral detox period.  His SpermGrandHag was high conflict and since the SpermIdiot rarely laid eyes on the kid even during his visitation time SpermGrandHag was who we had to deal with the most.

Give yourself a break and don't make the mistake of pursuing the BM/SM bond dream.  Focus on your marriage and putting that first. The kids in the mix are the top marital responsibility but not the priority.  Part of that responsibility is to develop as high quality a relationship as  you can with SD.  In my case that worked out well in my relationship with my SS-25. His mom and I met when he was 15mos old and married the week before he turned 2yo.  He asked me to adopt him when he was 22.

So, put as tight of a noose around BM's neck as  you and DH can and focus on your marriage and your family. Do what you can to mitigate and counter BM's toxic bullshit, and be a all in parent with your SD.  She will know who really has her back.  It may take a lifetime but kids are smart and with strong examples from you and your DH she and  your own daughter should be fine.

Good luck. 

Go ahead and accept BM doesn’t have to or want to meet you and it’s probably better for you anyway. The only relationship you can work on is that between you and your skid-not yours and bms or your SO’s and BMs. If you truly treat her well, the skid will eventually catch on that what BM is saying isn’t consistent to what she sees of you and it’ll discredit bm without you having to say/do anything.