The blazar 3C 454.3 experienced a major flare in November 2010, making it the brightest $\gamma$-ray source in the sky of the Fermi-LAT observatory. Motivated by the $3\sigma$ association of a $\gtrsim 290$ TeV muon neutrino IceCube170922A with an electromagnetic flare in TXS 0506+056 and noting that 3C 454.3 was up to $\sim 100$ times brighter than TXS 0506+056, we enquire what level of the neutrino flux we could expect from the brightest blazar flare of 3C 454.3. We construct a neutrino light curve of 3C 454.3 and estimate the expected neutrino yield at energies $\geq 100$ TeV for 3C 454.3 to be up to $6 \times10^{-3}$ muon neutrinos per year. We extrapolate our model findings to the light curves of all Fermi-LAT flat-spectrum radio quasars. We find that next-generation neutrino telescopes are expected to detect approximately one multimessenger ($\gamma + \nu_{\mu}$) flare per year from bright blazars with neutrino peak energy in the hundreds TeV - hundreds PeV energy range and show that the electromagnetic flare peak can precede the neutrino arrival by months to years. Finally, we cross-match the IceCube very-high-energy neutrino alerts from the IceCat-1 catalogue and Fermi-LAT light curves and find a few neutrinos spatially associated with bright blazars lagging behind their major $\gamma$-ray flares by $10^{2}-1.5\times 10^{3}$ d.