Let $α$ be a Steinhaus random multiplicative function. For a wide class of multiplicative functions $f$ we construct a multiplicative chaos measure arising from the Dirichlet series of $αf$, in the whole $L^1$-regime. Our method does not rely on the thick point approach or Gaussian approximation, and uses a modified second moment method with the help of an approximate Girsanov theorem. We also employ the idea of weak convergence in $L^r$ to show that the limiting measure is independent of the choice of the approximation schemes, and this may be seen as a non-Gaussian analogue of Shamov's characterisation of multiplicative chaos. Our class of $f$-s consists of those for which the mean value of $|f(p)|^2$ lies in $(0,1)$. In particular, it includes the indicator of sums of two squares. As an application of our construction, we establish a generalised central limit theorem for the (normalised) sums of $αf$, with random variance determined by the total mass of our measure.