In this paper, we are concerned with the existence and uniqueness of solutions for a class of three-point boundary value problems involving coupled differential systems with Atangana-Baleanu-Caputo fractional derivatives. The research employs upper-lower solution techniques combined with monotone iteration methods to establish sufficient conditions for the existence and uniqueness of maximal and minimal solutions. Furthermore, by constructing Green's functions and analyzing their properties, we rigorously prove the existence and uniqueness of extremal solutions while providing precise error estimates. To demonstrate the practical significance of our theoretical findings, we present a concrete application case that validates the system's effectiveness in real-world scenarios. Numerical simulations are also conducted to provide additional verification of the theoretical analysis, confirming the reliability of our proposed approach.