> What I have specifically is a class that provides a base set of operations and a subclass that lets you do some modifications

If you actually do this with public inheritance then you possibly break const correctness.

Having to separate types for the immutable and mutable data structures with appropriate conversions is a sound idea though.

Your object isn't actually immutable, it just doesn't let you directly mutate it. That's a really important distinction.

That's a common pattern. In C++, rather than using a subclass for the mutator, use a different (friend) class. The semantics for implicit conversion between two different classes will allow you to accomplish what you are asking for with enforcing no changes from one interface but not the other.