tldr; From my experience: Don't worry. Your daughter will be fine, and her world won't be upturned as long as no one around her acts like it is a big deal or acts like her world should be upturned. Keep it simple and let her ask the questions she cares about. I think you will find that it isn't as big of a change for her as it is for you, so don't make it out to be a big deal. She will follow your lead on that point in particular.
Kids really don't need more than a kids answer because all they will understand is a kids answer. Your daughter has no concept of "biological dad" or "real dad". Her current dad is her dad because she calls him that and because (presumably) he loves her, takes care of her, and spends time with her: i.e. he acts like her dad. As long as he continues doing that he will never stop being her dad, and her relationship with him will not change because someone else shows up, even if she calls the new guy "dad" too.
If you decide to have her call the new guy "dad" too, then I would just keep the introduction at a 4 year old level. "Meet your other dad: he is the dad who helped make you, and your other dad is the dad who helped raise you". Although every kid is different, your daughter will likely be plenty satisfied with that answer, and if she does have any followup questions stick to 4 year old answers.
Even very precocious kids are still just kids. In all likelihood the larger concepts you are worried about won't even occur to her. The idea is to give her a foundation for her understanding now and fill in the details as she grows and can understand more. As she grows older she may ask for more "details" herself (or as you think she is old enough you can share more details with her), and she should absorb the new information with little trouble.
My wife was a single mom when we got married: her son (now our son) was about two and a half. He's 9 now. He remembers when we first started dating. He remembers when we got married. He remembers when I adopted him. At some point in time he started calling me dad, but (now) it never occurs to him that I should be anything else. He's also a very smart kid, and we decided to tell him that he has "another" dad out there when he was 4 or 5. That revelation didn't cause any trouble for him. Relevant to you, it had never even occurred to him that he would have another dad out there. Even though he "knew" that I only showed up after he was born, he had no concept of a "biological" father. In his head the thing that made me his dad was the fact I was there as his dad every day. Biology wasn't a consideration.
His understanding of what this means has grown over time. He knows about sperm and eggs now, and understands that his "genes" came from his other dad. He understands what that means enough that he chuckles when people say we look alike. We've also told him that he has some half-siblings out there, and that just became one more detail that he rarely thinks about as he goes about his 9-year-old life.
Obviously every kid is different, but my own experience suggests you have nothing to worry about. Remember that 4 year olds really just need 4 year old answers. If her biological father starts spending time with her regularly, then that will definitely be a good thing. Having another guy in her life to love her, care for her, and look out for her will only ever work out for her good in the long run. So hopefully, the biological father will make a great dad too.
I wanted to primarily address your concerns over how to break the news to your daughter, but I think it is reasonable to also mention the behind-the-scenes legal implications of your question. Important qualifier: I am not a lawyer. Always seek professional advice. Your mileage will vary depending on your jurisdiction, but I can tell you what matters for the jurisdiction I am most familiar with: the state of Florida, in the good ol' USA.
The family law "division" of the judicial branch for the state of Florida works from the principle that children are best off when both biological parents are involved. As a result, a father who sues for custody in such a case has a good chance (a very broad, non-legal term) of being assigned at least partial custody. Certainly, nothing you have said would indicate any reason why the courts wouldn't grant him at least partial custody.
Of course such a process would be very expensive for the father: both in terms of legal fees and child support. If a father were to embark upon such a path, the state of Florida would also assign him child support payments so long as the mother was the "active custodian" for more than half of the time. Given that he already has two other kids (and is probably paying child support for at least one of them), it is probably unlikely that the father in question would attempt such a thing. It's an issue worth being aware of though. Again, if you are worried, seek professional advice specific to your jurisdiction.
To be clear, the issue is that whether or not to allow him into your daughter's life may not be entirely up to you. I know that thought can be scary (it use to scare the crap out of my wife before I adopted her son), but it is something to be aware of. In an ideal world, he will genuinely love and care for your mutual daughter, and so you will be able to let him in her life freely to the extent that he earns your trust.