Colloidal quantum dots are robust and flexible single photon emitters for room-temperature applications, but their purity is strongly reduced at high pump powers, due to multiexcitonic emission which cannot be easily filtered due to the photo-luminescence spectral broadening at room temperature. Giant-shell quantum dots feature a large blueshift of the biexciton spectrum due to electron-hole wave function engineering and piezoelectric charge separation, which can be exploited for spectral separation of the single exciton from the multiexciton emission. Here, by spectral filtering, we show that we can recover an excellent single-photon emission, with g2(0)<0.05 (resolution limited), even at high pump powers above saturation of the exciton emission. The bright and pure single-photon generation shown here has important applications in quantum information technology and random-number generation