The distribution of partial sums of random multiplicative functions with a large prime factor

For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums $$\frac{{(\log \log x)}^{1\mathrm{/}4}}{\sqrt{x}}\sum _{\begin{array}{c}n\le x\\ P(n)>\sqrt{x}\end{array}}f(n),$$ where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.