$\newcommand{\si}{\sigma}\newcommand{\Si}{\Sigma}$The answer is: in general, no.
Indeed, let $C:=\mathcal C=(-\infty,1)\times(-\infty,\infty)$, $v=(0,0)$, $f:=\phi=(1,0)$, and $\Si=\begin{pmatrix}\si^2&0\\0&1\end{pmatrix}$, with $\si\to\infty$. Then $$P(u\in C)\ge P\big(u\in(-\infty,0)\times(-\infty,\infty)\big)=1/2$$ and \begin{equation} P(u\in C,f\cdot u>f\cdot v)=P(1>f\cdot u>0)\to0, \end{equation} so that $P(f\cdot u>f\cdot v\,|\,u\in C)\to0$.