5yo DD has been super strong willed since birth. I don't know how else to say this, and I know it sounds awful, but she constantly lies, and I swear she is highly manipulative. Some of it is minor, like she swears up and down she has brushed her teeth, "Mom I promise I promise! I'm not lying! I brushed my teeth!", then I go upstairs and her toothbrush is totally dry. But she lies about being sick a LOT to try and get medicine (she likes the taste) or as a excuse to not do something she doesn't want to do. I might in all honesty say, "hmm, if you are having headaches every day we should probably go see the doctor" and she immediately back pedals and screams that she hates the doctor (why?) and doesn't want to go and then eventually admits she is lying. I feel like I've screwed up her whole relationship with going to the doctor now, but damn I DO want to take her to get checked if she's really having headaches(!).
She tells us there is one girl at school that is mean to her, but then when I tell her that I feel like we should speak to the teacher because the things she is saying aren't ok, she again freaks, saying, "It's fine, don't tell the teacher" or "maybe I just don't remember" etc. then admits to making it up. She often purposely mischaracterizes situations for attention or to avoid getting trouble.
I get that it's an attention thing, but she gets a lot of attention at home and other places. She is incredibly cute (very short for her age, hysterical, and has a really fun personality) so she easily attracts attention and is often the teachers pet. To be honest she demands as much attention from everyone around her as they can possibly give.
She doesn't push or hit, but she has trouble giving people personal space because she wants to be all up in their grill ALL the time. My chill 10 YO is in tears or just super angry most afternoons because she needs alone/quiet time, and younger DD can't leave her alone for 5 minutes, even when asked nicely and politely MANY times before hand. By the end of the day we are all exhausted, and I will admit sometimes I lose patience which I know doesn't help anything. She has started yelling at me and being downright disrespectful, so I'm trying soooo hard to not yell anymore.
We do timeouts, consequences like no screen time etc., reward charts, making a HUGE deal of things when she makes good choices/tells the truth, all with minimal success. With the lying, we've gone from explaining what happens when someone can't tell if you're telling the truth or not (Boy who cried wolf type convo), to explaining how dangerous it is to lie about sickness or injury (what if she really IS having headaches

), to explaining how I need to be able to trust her because we need to be a team, and on and on. None of it is clicking at all.
Now within the past 2 weeks, when she gets in trouble or we talk to her, she calls herself mean and stupid and says that we hate her, which is so concerning and it's breaking my heart. She compares herself to her sister and says she wishes she was her(who is a total people pleaser and NEVER gets in trouble). But hand to God sometimes when she says this stuff she is fake crying - trying to get a reaction out of us. I try to end each day discussing several things I think she did well that day and why I'm proud of her and that I will love her no matter what. She's clearly calling out for some sort of help that we aren't giving her. I'm honestly feeling like a total failure as a parent and I don't know where to go from here. Open to therapist and other suggestions.