This problem has been created by your BF rewarding bad behaviour - as you seem to be aware.  When he says "guess I'm a bad dad" - yes you damn well are, should be the response! He needs it explained to him that not only is he doing his daughter a gross disservice by his lax parenting, but his relationship with her as well, in the long term.  He will turn her into an unlikeable tyrant who will not be able to maintain friendships at school, and later in life will be unemployable and probably live with him or his exW forever!   You seem to have a lot of the day to day care of his daughter, but none of the authority to set parenting rules as you would wish them to be.  This is a bad situation and not sustainable in the long term.  Something needs to change. 

We had this struggle while raising my SS-30.  His mom and I married the week before he turned 2yo.

In our case it was SpermGrandHag that was the instigator of the "He is not your REAL dad, he is only  your StepDad.  You can't call him dad." bullshit.

This is how I dealt with it.

I was the first person my SS ever called Dad(dy).  His mom and I started dating when he was 15mos old.  No one told him to, he chose to call me daddy.  Prior to tagging me as Daddy, he would point at a pic on the wall of the Spermidiot when anyone was discussing "Daddy".  After several months of us dating, he pointed to me and said "Daddy". I have been his Dad(dy) ever since.

A few times over the visitation years he would come home from SpermLand visitation perplexed and spouting SpermGrandHag drivel about me not being his dad and how he could not call me dad. Which evolved into SpermGrandHag hammering him during visitaiton about me not being his REAL dad.

So.... We had this talk.

Rags:  Son, we have talked about how a Bio-Dad is the dad that made you with your mom.

A StepDad is the the dad that is married to  your mom and who loves  you and your mom very much.   

What a REAL dad is, is the dad that goes to work every day and works hard to provide a nice home for the family to live in, a safe neighborhood for you to live in, good schools for you to learn at, safe and reliable cars for the family to use, healthy food for the family to eat, teaches to use the toilet, to read, to tie your shoes, ride your bike, helps you with your home work, coaches your Swim Team, umpires your baseball league, makes sure the family can take nice vacations, and who loves you and your mom very much.

SS: Dad, a StepDad sounds like a REAL dad to me.  Can we go play now.

Later in the visitation years SS came home from a SpermClan visitation with "Dad, (SpermGrandHag) says I can't call you dad any more.

Rags:  Son. you started calling me dad before you were 18mos old.   So, if you no longer want to call me dad you have a choice. You can call me Mr. (Lastname).  I do not allow children to call my by my first name and I am not your uncle so you will not call me Uncle (Firstname).  So, your call. Will I be Dad or will I be Mr. (LastName).

SS:  I think I will stick with dad. That is who you are.

Adults who allow kids to call them by whatever the kid wants to call them by is not how life is in Rags' world. Mr, (LastName), Uncle (FirstName), or Dad. Depending on the kid.  They don't get to decide. They call me what I tell them to call me or they just refer to me as Sir. 

Period. Dot.

That is how I was raised to refer to adults. That is how I am referred to by children.

You decide how you will be addressed. It is up to you and only you.  If you choose to work out an alternative with the Skid, great. But it is YOUR call.

The culmonation of this topic was when SS asked me to adopt him when he was 22yo.  We made that happen. Now he carries our family name and wears it on his USAF uniform.  I am a very proud dad.  Make that... a very proud REAL dad.

IMO of course.

As for your lippy 4.Xyo SD..... A 4yo gets a stinging swat to the rump and their nose planted in a corner to  hold the walls togethe runtil you get tired when they get lippy. Lather..... rinse.... repeat.

Why would you or any other adult tolerate this crap from and ill behaved ill raised 4yo (any age for that matter)  failed family breeding experiment?

spermdonor.  If he won't immediately extricate his head from his own ass, man up, stop being a failed father, failed partner, and failed man, then leave and have your baby near your family, hopefully in another state, and protect yourself and your own child from the shallow and polluted end of it's gene pool.

Grrrrr!

Good luck.

Oh, my DH has used "I guess I'm just a bad parent then" or "I guess I don't know how to parent do I?" when we got into discussion of SDs behavior and what to do about it. He would NOT hear my side of it nor would he ever do what I suggested - I don't think he ever, ever has done this. Which is why I left it all to him. It got easier as the SDs got older, don't know what I would have done if the SDs were under age 5 (well, come to think of it if they were I wouldn't have dated him because my cut off was a guy with kids minimum age 5 and up only).  

If he wants/needs you to have the responsibility of being in a parent role, with more time spent with you then he MUST allow you the parental authority to do so. This means HE needs to agree with your parenting decsions and style OR he must have a realistic discussion with you without dismissing your perspective and concerns. Otherwise you will not be able to ever become anything but a babysitter. And it will get worse. 

Ask him if he'd hound a teacher like that. Point out that if you were a nanny, to start with you"d be paid for your time AND you'd walk out the first time he undermined you. 

Oh bugger.., the next time he undermines you, get up, leave the room and tell him you will no longer be his unpaid nanny and he has 8hours to find a new one. Set your alarm before his and be sure to leave the house before him. Do not answer his calls. Not your problem. If he can't respect you, he doesn't get respect, 

The same thing happened to me. When my DH's 3 daughters were young, the younger 2 wanted to call me mom and make me mother's day gifts along with gifts for their biomom. BM got wind of this and suddenly the kids wouldn't hug me, wouldn't have anything to do with me. It was wierd, so DH asked them what was going on, and they finally said that BM told them I was "a bad person and not their mom."  Even though we address this with them, they still felt really uncomfortable around me for about a month - until I showed them through my actions that I was NOT a bad person. Then they were really just confused about who to believe. I finally disenganged because it was too hurtful to put all of that energy into helping them, caring for them, and then have them hurl insults at me when I had to make them do their chores or go to bed on time. These insults clearly came from biomom. The oldest SD was already messed up due to biomom's PASing her and making her act as a sheepdog for her younger siblings. When they started to have some affection for me, oldest SD would shut it down, acting as proxy for BM when they were at our house. It was sad, and it still makes me sad, seeing how these kids turned out because I had to disengage. BM can really ruin a kids life that way.

I would recommend disengaging from SD until DH can stand up and intervene in BM's BS. However, even if DH does that, BM's influence is so strong that you may never be able to have a relationship with this little girl. Its sad for her, and for you - but you are not responsible for BM's attitude or this child.