Flare loops form an integral part of eruptive events, being detected in the range of temperatures from X-rays down to cool chromospheric-like plasmas. While the hot loops are routinely observed by the Solar Dynamics Observatory’s Atmospheric Imaging Assembly, cool loops seen off-limb are rare. In this paper we employ unique observations of the SOL2017-09-10T16:06 X8.2-class flare which produced an extended arcade of loops. The Swedish 1 m Solar Telescope made a series of spectral images of the cool off-limb loops in the Ca II 8542 Å and the hydrogen H-beta lines. Our focus is on the loop apices. Non-LTE spectral inversion (non-LTE; i.e., departures from LTE) is achieved through the construction of extended grids of models covering a realistic range of plasma parameters. The Multilevel Accelerated Lambda Iterations (MALI) code solves the non-LTE radiative-transfer problem in a 1D externally illuminated slab, approximating the studied loop segment. Inversion of the Ca II 8542 Å and H-beta lines yields two similar solutions, both indicating high electron densities around 2x10^12 cm^(-3) and relatively large microturbulence around 25 km/s. These are in reasonable agreement with other independent studies of the same or similar events. In particular, the high electron densities in the range 10^12 - 10^13 cm^(-3) are consistent with those derived from the Solar Dynamics Observatory’s Helioseismic and Magnetic Imager white-light observations and they are also required to explain SST/CHROMIS continuum observations in the wide-band channel centered at 4845.5 Å.