Is there any meromorphic matrix $F$ on $\mathbb{C}^*$, with a non trivial pole at $0$ and with values in $O_2(\mathbb{C})$ ?
My guess is that a such matrix doesn't exist, but i'm not sure.
I've tried to find contradiction to : $f_1^2 +f_2^2 =1$ with $f_1,f_2$ meromorphic functions on $\mathbb{C}$.
Try $$g(z)=i\frac{z+1}{z-1},\qquad f_1(z) = \frac{1-g(z)^2}{1+g(z)^2},\qquad f_2(z) = \frac{2g(z)}{1+g(z)^2}$$ $$ F(z)=\pmatrix{f_1(z)&f_2(z)\\-f_2(z)&f_1(z)}$$
Explanation: check that $f_1^2+f_2^2=1$ (the rational parametrization of the circle),
$g,f_1,f_2$ are meromorphic everywhere including at $\infty$,
$g$ has a pole at $1$ but $f_1,f_2$ are holomorphic there,
$f_1,f_2$ have a pole whenever $g(z_0)=\pm i$ which happens only at $0$ and $\infty$.