$\newcommand{\si}{\sigma}\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}$The answer is: in general, no -- even for convex $\mathcal C$.
Indeed, let $C:=\mathcal C=(-\infty,1)\times\R$, $v=(0,0)$, $f:=\phi=(1,0)$, and $\Si=\begin{pmatrix}\si^2&0\\0&1\end{pmatrix}$, with $\si\to\infty$. Then $$P(u\in C)\ge P\big(u\in(-\infty,0)\times\R\big)=1/2$$ and \begin{equation} P(u\in C,f\cdot u>f\cdot v)=P(1>f\cdot u>0)\to0, \end{equation} so that $P(f\cdot u>f\cdot v\,|\,u\in C)\to0$.
This example is written for dimension $2$, but can be easily modified for any dimension $\ge1$.