Is argmin of a sum of two functions equal to sum of each argmin?
No. Consider $f(x)=-x,g(x)=(1+x^2)^2,h(x)=1+x^2$. Then $$\bar \theta=\min_{x\in\mathbb{R}}\left\{\frac{g(x)}{h(x)}=1+x^2\right\}= 1,\\ \arg\min_{x\in\mathbb{R} }\left\{f(x)+\frac{g(x)}{h(x)}=x^2-x+1\right\}=\frac{1}{2},\\ \arg\min_{x\in\mathbb{R} }\left\{f(x)+g(x)-\bar\theta h(x)=x^4+x^2-x\right\}\neq \frac{1}{2}.\\ $$