I think the LMIII is slightly taller and slightly deeper than the II. Depends how much breathing room is in the bag
Contact: OSP Cases And ask if available, or if they will make one for you. For reference, here's what it looks like. https://www.amazon.com/OSP-Amplifier-Markbass-Little-ATA-MARKB-LM800/dp/B009YWQ2QG I bought one for my LMT-800 from an E-Bay listing several years back. Sorta pricey, but- The MB soft case is the ticket most of the time, but for longer-distance road travel, a flight-type case like this is the way to go.
@Harry Wilson - I use this: Gator CasesG-MIXERBAG-1212 Padded Nylon Mixer/Equipment Bag (12.0 x 12.0 x 5.5
Thread drift: I can recall a time when those "silver briefcases" were the exclusive (and rarified) province of the Zero-Halliburton company, and they conveyed an extremely high standard of protection/durability. Not sure which I find more distressing now: That one can buy their cases in truck stops, or that the name "Halliburton" has become associated more with recent activities in the geo-political sphere than with high quality cases. Both would seem to sully the brand. Back on track: For local gigs I carry my mini-head in a padded nylon bag similar to the Gator pictured above; if we're travelling more than ~50 miles I leave the head in that bag and put the whole thing into a foam-lined Anvil plywood road case.