We present a multiwavelength analysis of 28 of the most luminous low-redshift narrow-line, ultra-hard X-ray-selected active galactic nuclei (AGN) drawn from the 70-month Swift/BAT all-sky survey, with bolometric luminosities of log(Lbol/erg s-1) ≳ 45.25. The broad goal of our study is to determine whether these objects have any distinctive properties, potentially setting them aside from lower luminosity obscured AGN in the local Universe. Our analysis relies on the first data release of the BAT AGN Spectroscopic Survey (BASS/DR1) and on dedicated observations with the VLT, Palomar, and Keck observatories. We find that the vast majority of our sources agree with commonly used AGN selection criteria which are based on emission line ratios and on mid-infrared colours. Our AGN are pre-dominantly hosted in massive galaxies (9.8 ≲ log (M∗/M) ≲ 11.7); based on visual inspection of archival optical images, they appear to be mostly ellipticals. Otherwise, they do not have distinctive properties. Their radio luminosities, determined from publicly available survey data, show a large spread of almost four orders of magnitude - much broader than what is found for lower X-ray luminosity obscured AGN in BASS. Moreover, our sample shows no preferred combination of black hole masses (MBH) and/or Eddington ratio (λEdd), covering 7.5 ≲ log (MBH/M) ≲ 10.3 and 0.01 ≲ λEdd ≲ 1. Based on the distribution of our sources in the λEdd-NH plane, we conclude that our sample is consistent with a scenario where the amount of obscuring material along the line of sight is determined by radiation pressure exerted by the AGN on the dusty circumnuclear gas.