Photometric and spectroscopic analyses have shown that the Galactic bulge cluster Terzan 5 hosts several populations with different metallicities and ages that manifest as a double red horizontal branch (HB). A recent investigation of the massive bulge cluster NGC 6569 revealed a similar, though less extended, HB luminosity split, but little is known about the cluster's detailed chemical composition. Therefore, we have used high-resolution spectra from the Magellan-M2FS and VLT-FLAMES spectrographs to investigate the chemical compositions and radial velocity distributions of red giant branch and HB stars in NGC 6569. We found the cluster to have a mean heliocentric radial velocity of -48.8 km/s ({sigma}=5.3 km/s; 148 stars) and <[Fe/H]>=-0.87 dex (19 stars), but the cluster's 0.05 dex [Fe/H] dispersion precludes a significant metallicity spread. NGC 6569 exhibits light- and heavy-element distributions that are common among old bulge/inner Galaxy globular clusters, including clear (anti)correlations between [O/Fe], [Na/Fe], and [Al/Fe]. The light-element data suggest that NGC 6569 may be composed of at least two distinct populations, and the cluster's low <[La/Eu]>=-0.11 dex indicates significant pollution with r-process material. We confirm that both HBs contain cluster members, but metallicity and light-element variations are largely ruled out as sources for the luminosity difference. However, He mass fraction differences as small as {Delta}Y~0.02 cannot be ruled out and may be sufficient to reproduce the double HB.