Well, more like I’m looking for something that gives you a const that you can’t get rid of. You could use a factory method but it’d be nicer if I could have a constructor that did it.
That's just not how C++ is designed. If you want const, you stick const on the member variables or declare member getter functions const. Then the language checks your work when you initialize an object into a constant variable.
This allows you to force an object to be immutable at a compiler enforced level regardless of how the calling code tries to initialize it.
Since soft const constraint is not part of the type, perhaps you could pass it to a lambda function that treats it as a const and another lambda function that treats it as mutable. A sort of const region and mutable region.