Trivial source characters in blocks of domestic representation type

Let $G$ be a finite group of even order, let $k$ be an algebraically closed field of characteristic $2$, and let $B$ be a block of the group algebra $kG$ which is of domestic representation type. Up to splendid Morita equivalence, precisely three cases can occur: $k{V}_4$, $k{\mathfrak{A}}_4$ and the principal block of $k{\mathfrak{A}}_5$. In each case, given the character values of the ordinary irreducible characters of $B$, we determine the ordinary characters of all trivial source $B$-modules.