No Campag and Shimano cassettes are never compatible/interchangeable. You can stick a 8, 9 or 10 speed Shimano cassette on any modern 8, 9 or 10 speed Shimano hub. You can not stick a Campag 9, 10 or 11 speed Campag cassette on a Campag 8 speed hub.
I'm with you: Shimano hubs use the same splines across 8, 9 & 10; Campag 8 speed hubs are different from 9, 10 & 11 speed hubs.
It is impossible to see clearly from the photo, with vintage stuff there were indeed all sorts but it doesn't look a million miles from the "standard" Shimano/Campag one attached below or does it? If it looks like the one in the photo at the bottom then you will have to stick with vintage French shifters I am afraid.
Yes, it does look like the one in the photo, in that it doesn't have any "flats" on the sides (which the boss in the diagram does. I'm after a pair of Simplex LJ shifters, so I'm hopeful they will fit.
One other potential issue indicated by your Huret shifter, being French, is some as late as 80's need different threads which the typical Shimano/Campag bolts won't fit (but you
might be able to use existing bolts). Is the frame French too? Separately can you work out, since left side fits the right side doesn't, what is causing the problem on the right?
It's a 1989 Peugeot but the threads don't appear to be the problem, as (without the lever) I can screw the bolt all the way in. I now think the problem is the lack of flats on the boss, which mean the lever doesn't sit far enough down the boss.
If indexing is not important to you it makes no difference which wheel you use except you could have more speeds if you want with the Shimano wheel by changing cassette/chain. If indexing is important to you and bosses are "modern" conceivably a Shimano 6/7 speed downtube indexed shifter will index 6/7 of any of the adjacent sprockets of a Shimano 8/9 speed cassette* respectively on your Shimano wheel (see pink boxes in the 3rd table
here)
with your "new" Campag rear mech. If it looks/sounds confusing, it is.
* if I do this myself I would rearrange the sprockets
I'm not bothered about indexing and will be using the Shimano wheel on the Pug and the 8 speed Campag on the TT bike I'm building up. I'll put my "new" 9 speed derailleur to one side until I can source a 9 speed hub: probably best to get what I have running properly, before tackling a new project!