FH,
Speculating here but I think that the first port of call is to see if the problem is one of just another loading restriction.
Consider if the forward limit has a comparatively shallow slope (this may occur toward the max weight or if the various limitations result in a visor-shaped spike in the graph .. not seen all that often .. depends on how the OEM worked up the limitations). If the loading puts the aircraft at the forward limit in this case, then an aft movement of CG will permit an increase in GW as one moves along the limiting line.
Hence my interest in the actual envelope (the TCDS doesn't list the details .. merely refers one to the AFM) and the GW/CG intersection where the problem has been noted.