Cherenkov Telescope Array Observatory (CTAO) is the next-generation ground-based $\gamma$-ray observatory operating in the energy range from $20\, \mathrm{GeV}$ up to $300\, \mathrm{TeV}$, with two sites in La Palma (Spain) and Paranal (Chile). It will consist of telescopes of three sizes, covering different parts of the large energy range. We report on the performance of Large-Sized Telescope prototype (LST-1) in the detection and characterization of extragalactic $\gamma$-ray sources, with a focus on the reconstructed $\gamma$-ray spectra and variability of classical bright BL Lacertae objects, which were observed during the early commissioning phase of the instrument. LST-1 data from known bright $\gamma$-ray blazars – Markarian 421, Markarian 501, 1ES 1959+650, 1ES 0647+250, and PG 1553 + 113 – were collected between 2020 July 10, and 2022 May 23, covering a zenith angle range of 4$^\circ$ to 57$^\circ$. The reconstructed light curves were analysed using a Bayesian block algorithm to distinguish the different activity phases of each blazar. Simultaneous Fermi-LAT data were utilized to reconstruct the broad-band $\gamma$-ray spectra for the sources during each activity phase. High-level reconstructed data in a format compatible with gammapy are provided together with measured light curves and spectral energy distributions (SEDs) for several bright blazars and an interpretation of the observed variability in long and short time-scales. Simulations of historical flares are generated to evaluate the sensitivity of LST-1. This work represents the first milestone in monitoring bright BL Lacertae objects with a CTAO telescope.