In this paper, we are concerned with the following mixed order conformally invariant system with exponential Hartree nonlinearity and cubic nonlinearity: \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{\frac{1}{2}} u(x)=\left( \frac{1}{|x|^2}*e^{2pv(x)}\right) e^{pv(x)},~~~& x\in \mathbb {R}^3,\\ ~~~~~~~~~~(-\Delta )^{\frac{3}{2}} v(x)= u^{3}(x),~~~& x\in \mathbb {R}^3, \end{array}\right. } \end{aligned}\) where \(p>0\) , \(u\ge 0\) , v may change sign and u satisfies the finite total curvature condition \(\int _{\mathbb {R}^3} u^3(x)\textrm{d}x<+\infty \) . Under extremely mild assumptions, we prove that, the classical solution (u, v) must take the unique form: \( u(x)=\frac{2^{\frac{4}{3}}p^{-\frac{1}{3}}\mu }{1+\mu ^2|x-x_0|^2},\qquad v(x)=\frac{1}{p}\ln \Bigg [\frac{\left( \frac{2^{\frac{7}{3}}p^{-\frac{1}{3}}\pi ^2}{I^2(1)}\right) ^{\frac{1}{3}}{\mu }}{1+\mu ^2|x-x_0|^2}\Bigg ] \) for some \(\mu >0\) and \({{x}}_0\in \mathbb {R}^{3}\) , where \(I(1):=\frac{\pi ^{\frac{3}{2}}\Gamma (\frac{1}{2})}{\Gamma (2)}\) .
