We present $Spitzer$ IRS 5--14 $mu$m spectra and 16 $mu$m and 22 $mu$m photometry of the T2.5 companion to the $sim$300 Myr-old G0V star HN Peg. We incorporate previous 0.8--5 $mu$m observations to obtain the most comprehensive spectral energy distribution of an intermediate-gravity L/T-transition dwarf which, together with an accurate Gaia EDR3 parallax of the primary, enable us to derive precise fundamental parameters. We find that young ($approx$0.1--0.3 Gyr) early-T dwarfs on average have $approx$140 K lower effective temperatures, $approx$20% larger radii, and similar bolometric luminosities compared to $gtrsim$1 Gyr-old field dwarfs with similar spectral types. Our accurate infrared spectrophotometry offers new detail at wavelengths where the dominant carbon-bearing molecules have their strongest transitions: at 3.4 $mu$m for methane and at 4.6 $mu$m for carbon monoxide. We assess the performance of various widely available photospheric models and find that models with condensates and/or clouds better reproduce the full SED of this moderately young early-T dwarf. However, cloud-free models incorporating a more general convective instability treatment reproduce at least the low-resolution near-IR spectrum similarly well. Our analysis of $Rapprox2300$ $J$-band spectra shows that the near-infrared potassium absorption lines in HN Peg B have similar strengths to those seen in both younger and older T2-T3 dwarfs. We conclude that while alkali lines are well-established as surface gravity indicators for L-type or warmer stars, they are insensitive to surface gravity in early-T dwarfs
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