Oh my. This is complicated. First: Your fiancé is absolutely wrong: This is not normal boy behavior. This little boy is in emotional turmoil and is expressing it in the only way a three year old can — behaviorally. From his point of view, his mother abandoned him and you went and got a son of your own. His father is abandoning him too — by not giving him the structure and guidance that a little boy needs.
I realize it’s hard to love a kid who is this out of control. But that’s exactly what he needs. He is testing to see if anyone will love him unconditionally, regardless of his behavior. He pushes every limit to see whether you and his dad will reject him as his mother did.
Loving him does not mean tolerating this behavior. Loving him actually means that you and his dad have to do the hard work of getting on the same page and sticking to solid limits and clear consequences, while at the same time catching him every time he does something right. You will probably need a family therapist to help the two of you sort out how to accomplish this and how to be supportive of each other while you go through a very difficult transition.
The good news is that the boy is only 3. A thoughtful and consistent intervention now is likely to start to pay off within months.
I wish you well.