How to deal with “seasonal” parents?

So I’ve been in my SD (10) life since she was 1 yo. In the beginning the BM was so so petty. Like calling cops on us to come get our SD (3) bc we didn’t feel she was safe around her then-boyfriend (who was also on the phone yelling at my SO always trying to fight him over stuff she started, and who eventually DID turn out to be a bad guy). It was back and forth a while with her giving her to us bc she didn’t have a place or was moving or whatever, then months later wanting her back, then giving her back again. And so on… Well since she was 5 she has permanently been with us. We put her through school etc. Though she would always like to bring up that its bc “she lets us keep her” bc she’s the “custodial parent” since BM and SO weren’t married and they never went to court to establish a formal custody agreement. Now fast forward to 10yo. BM moves constantly and never seems to prioritize SD. Doesn’t come to any of SD school events and very rarely games. SD always is the one to initiate contact during the week and has even gotten to the point to where she tells her friends and therapist that her mom lies a lot and usually doesn’t pick her up when she says she will. And has asked me “why her mom doesn’t want her?” (Bc she lived 30 minutes away and still would never see her) Side note: we pay $400/mo in Child Support to BM and we pay for all SD insurance and all necessities since SD lives with us. Just really frustrated because we pretty much have to walk on eggshells so she doesn’t just come take SD away, but kinda pissed bc SD deserves so much better than this. My heart breaks for her because she tries so hard.