Using femtosecond transient absorption (fs-TA), we investigate the hot exciton relaxation dynamics in strongly confined lead iodide perovskite nanoplatelets (NPLs). The large quantum and dielectric confinement leads to discrete excitonic transitions and strong Stark features in the TA spectra. This prevents the use of conventional relaxation analysis methods extracting the carrier temperature or measuring the buildup of the band-edge bleaching. Instead, we show that the TA spectral line shape near the band-edge reflects the state of the system, which can be used to probe the exciton cooling dynamics. The ultrafast hot exciton relaxation in one- to three- monolayer-thick NPLs confirms the absence of intrinsic phonon bottleneck. However, excitation fluence-dependent measurements reveal a hot phonon bottleneck effect, which is found to be independent of the nature of the internal cations but strongly affected by the ligands and/or sample surface state. Together, these results suggest a role of the surface ligands in the cooling process.