FALSE: modular representations are determined by ordinary characters

Let $V$ be the $2$-dimensional $k{C}_p$-module with generator acting by $I+N=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, and let $W=k\oplus k$ be the direct sum of two trivial modules. Since $N^2=0$ and $\mathrm{char}k=p$, $(I+N{)}^p=I$, so this is a representation of $C_p$. The modules are not isomorphic: the generator acts nontrivially on $V$ and trivially on $W$.

For each group element $g^a$, the matrices are $(I+N{)}^a=I+aN$ on $V$ and $I$ on $W$, so their ordinary-style traces are both $2$ in $k$. The invariant line $k{e}_1\subset V$ and its quotient are both trivial, so $V$ and $W$ also have the same semisimplification. The only $p$-regular element of $C_p$ is the identity, where both Brauer characters, when defined through a splitting system, take the value $2$. These conclusions follow directly from the matrices and require no splitting-system hypothesis on $k$.

Thus ordinary-style character data do not recover the full modular representation; even Brauer characters cannot distinguish these two modules. The statement is false.