Impurity doping of low-dimensional semiconductors is an interesting route towards achieving control over carrier dynamics and energetics, e.g., to improve hot carrier extraction, or to obtain strongly Stokes shifted luminescence. Such studies remain, however, underexplored for the emerging family of III-V colloidal quantum dots (QDs). Here, we show through a detailed global analysis of multiresonant pump-probe spectroscopy that electron cooling in copper-doped InP quantum dot (QDs) proceeds on subpicosecond time scales. Conversely, hole localization on Cu dopants is remarkably slow (1.8 ps), yet still leads to very efficient subgap emission. Due to this slow hole localization, common Auger assisted pathways in electron cooling cannot be blocked by Cu doping III-V systems, in contrast with the case of II-VI QDs. Finally, we argue that the structural relaxation around the Cu dopants, estimated to impart a reorganization energy of 220 meV, most likely proceeds simultaneously with the localization itself leading to efficient luminescence.