You didn’t mention what the shoulder angle was but if it’s 35 or 40 degrees it is likely a 35 Short Ackley Magnum as you stated. I said “likely” because the dimensions of any one particular short Ackley Magnum can differ from others with the same name. After all, they are wildcats and the reamer could have cut a slightly different chamber, depending on the reamer maker and/or the specs provided by the owner. The 35 Short Ackley Magnum in my collection varies from yours in that it has a slightly larger diameter shoulder (less body taper) and a longer body (base to shoulder). But the fact that yours is made from 375 H&H brass lends considerable weight to it being a 35 Short Ackley Magnum. What is the headstamp BTW? That is a good clue to its age.
The 358 Norma Magnum has many dimensions very similar to yours and it could also be a custom variation of that cartridge. The Norma case has a 25 degree shoulder. But since Norma brass is available over the counter it would seem unlikely that a shooter would have opted to make brass from the full length 375. But never trust a wildcatter to do what makes sense.
A 35 Belted Newton is also strikingly similar in dimensions as is the 350 Womak Express.
If you could post a photo of the cartridge along side a well known such as the 30-06, it would help. Identifying someone else’s wildcat cartridge can be exasperating and there are times when a 100% ID is impossible. You could always call it a 350 Kchapa and be done with it.
(I’m nit-picking here, but the 375 H&H is a Belted case, not a Belted Rimless.)