In single-mode fiber (SMF), per-channel launch power is not a free design variable. The Kerr effect limits each channel's power through self-phase modulation (SPM), while cross-phase modulation (XPM), four-wave mixing (FWM), and inter-channel stimulated Raman scattering (ISRS) couple the launch powers of all channels to their quality of transmission, requiring joint optimization of the launch profile. Hollow-core fiber (HCF), whose Kerr coefficient is three to four orders of magnitude lower than that of silica, makes the per-channel powers nearly independent design variables constrained primarily by the shared amplifier-output budget. We develop a per-channel generalized signal-to-noise ratio (GSNR) model for amplified HCF links that includes amplified spontaneous emission (ASE) with a wavelength- and output-power-dependent erbium-doped fiber amplifier noise figure, intermodal interference, nonlinear distortion in EDFA pigtails, carbon-dioxide gas-line loss, and a wavelength-dependent transceiver ceiling. We also derive a sensitivity law that predicts when launch-power shaping is beneficial: the achievable gain is bounded by the ASE-noise fraction, is cancelled by Kerr nonlinearities near the SMF operating point, and remains positive in HCF. For 80-channel, 64-GBaud C-band links spanning 400-3200 km, per-channel optimization improves the worst-channel GSNR by up to 1.1 dB relative to flat launch, reduces cross-channel power sensitivity by one to two orders of magnitude compared with SMF, and achieves a given GSNR with approximately 11 dB less amplifier output power. Under a fixed amplifier-output budget, this advantage yields a 1.6-2.9-fold extension in worst-channel reach for GSNR targets of 20-23.4 dB, whereas the same optimization extends the reach of the nonlinearity-limited SMF link by at most 9%.