Once you get into lightweight 'proper' cello cases that genuinely offer protection to the instrument, as against styrofoam or semi-rigid, which I suppose you could say are maybe better than a good cello bag, the lightweight tag tends to come at quite a price. I have a Bam Shuttle with wheels which I think probably equates to the Classic nowadays, the case is excellent, but the wheels, whilst useful on a flat surface are not much good for anything which isn't a good pavement or road. However, I don't think I'd call it exactly a lightweight case. Bam do of course make a lightweight case, 2.9 kilos but at around £1000 it's pricey. My Bam is now around 20 years old, so the £300 I spent then was not wasted. I can carry it a fair way but wouldn't want to go for a hike with it ...
If you don't get your definitive answer from someone, perhaps it may be worth calling Caswells, who have a very wide range, and a good deal of expertise and experience. They may be able to advise you. They have good descriptions too of each case. https://www.caswells...&max_price=1696
To be honest when I was teaching cello I ended up mostly using a really good well padded cello bag, which survived without any damage whatsoever to the case or the cello, which is a nice 19C instrument, but not of great value. I did tend to overload it with music so it was a bit of a beast to back pack around. 5-6 kilos, or 3-4 kilos for the price difference, well, it's hard to justify the extra cost. My Bam violin case, with extra pockets and as my only hand luggage when loaded for a flight weighs around 12 kilos.
Much of it depends on the value of the instrument and just how much you want to spend.