We present an optical through sub-millimetre multi-wavelength study of two very low-luminosity Class I/Flat systems, Mayrit 1701117 and Mayrit 1082188, in the sigma Orionis cluster. We performed moderate resolution (R 1000) optical ( 0.4-0.9mu) spectroscopy with the TWIN spectrograph at the Calar Alto 3.5-m telescope. The spectra for both sources show prominent emission in accretion- and outflow-associated lines. The mean accretion rate measured from multiple line diagnostics is 6.4x10^{-10} Msun/yr for Mayrit 1701117, and 2.5x10^{-10} Msun/yr for Mayrit 1082188. The outflow mass loss rates for the two systems are similar and estimated to be 1x10^{-9} Msun/yr. The activity rates are within the range observed for low-mass Class I protostars. We obtained sub-millimetre continuum observations with the Submillimetre Common-User Bolometer Array (SCUBA-2) bolometer at the James Clerk Maxwell Telescope. Both objects are detected at a >5-sigma level in the SCUBA-2 850mu band. The bolometric luminosity of the targets as measured from the observed spectral energy distribution over 0.8-850mu is 0.18+/-0.04 Lsun for Mayrit 1701117, and 0.16+/-0.03 Lsun for Mayrit 1082188, and is in the very low-mass range. The total dust+gas mass derived from sub-millimetre fluxes is 36 M_Jup and 22 M_Jup for Mayrit 1701117 and Mayrit 1082188, respectively. There is the possibility that some of the envelope material might be dissipated by the strong outflows driven by these sources, resulting in a final mass of the system close to or below the sub-stellar limit. Given the membership of these objects in a relatively evolved cluster of 3 Myr of age, we consider an alternate formation mechanism in the context of the `hybrid' model of disk fragmentation, followed by ejection of a gaseous clump.