Is there a non Turing-complete model of computation whose halting problem is undecidable?
You can easily build artificial models which are not Turing complete but the halting problem for them is undecidable. E.g. take all TMs that do not halt on anything but $0$.
Regarding the statement:
You cannot disprove a statement that is not precise enough. Almost none of the words in the statement is well-defined (please provide the definition for them if this is not the case).