On the one hand, JS DOM objects are idl generated wrappers for C++. In theory we can generate more WASM friendly wrappers.
On the other, the C++ code implementing the API will be tightly coupled to the entire JS type system and runtime. Not just the concept of an object but every single design decision from primitives to generators to dynamic types to prototypical inheritance to error handling...
Also, I believe the C++ DOM implementation itself is pretty tightly integrated with javascript and it's memory management e.g. nodes have references into the managed heap to use JS objects directly like EventListeners and js functions.
Creating a new non-JS DOM API doesn't sound intractable to me... but browsers annihilate my assumptions so it's probably millions of hours of effort and close to a rewrite...