We study abelian fractional quantum Hall edges with $N$ co-propagating interacting channels and a local quantum point contact (QPC). An injected anyon generally does not acquire a universal statistical phase $\theta$, but fractionalizes into $N$ propagating components whose individual phases depend on interactions. Their sum, however, is protected and satisfies $\pi \delta =\theta (mod\pi )$, thereby defining a universal local scaling dimension $\delta$ at the QPC. This fractionalization distinguishes anyon species carrying the same charge and exchange phase, whose mutual phase is also split. We show that time-domain braiding is recovered when a single anyon species enters and when either all propagating modes have identical velocities, requiring fine tuning of the interactions, or for a local injection directly at the QPC. Exploiting a local anyonic time-domain exchange (ATE) relation between backscattered anyon and quasihole, a necessary consequence of time-domain braiding that remains valid beyond it, we propose minimal single-QPC protocols that isolate $\theta$ as a genuine exchange phase, distinct from its role in the scaling dimension, while simultaneously testing the ATE relation. The protocols rely on two nonequilibrium fluctuation-dissipation relations that connect the DC backscattering noise either to an integral of the DC current or to the phase of the admittance, even at low frequencies. Remarkably, for thermalized edges in the quantum regime, the admittance phase is identified with $\theta$ whenever $\delta >1/2$.