One thing I don’t quite like is that there would be terms t :: ty whose semantics has ⟦t⟧ ∉ ⟦ty⟧.
Other than that, I think I see what you did there. Instead of the categories Hask and DCPO, you are considering the categories KindedHask and KindedDCPO where morphisms are functions and objects are objects of Hask resp. DCPO tagged with either * or #, the kind. In DCPO, only domains with ⊥ may be tagged with *. Then you define a mapping Comp on objects of KindedDCPO such that it is Lift for every object tagged with # and the identity for every object tagged with *. The question then is whether we can make Comp an endofunctor or even a monad. If we could, then we can hope for ⟦_⟧ to be a functor from KindedHask to the Kleisli category of Comp.
Let us define the morphism part of the functor Comp. Let 𝜂 denote the unit of the Lift monad Lift x = x + ⊥. I see only one choice:
Comp (f: * -> *) = f
Comp (f: * -> #) = 𝜂 . f
Comp (f: # -> #) = Lift f
Comp (f: # -> *) = maybe ⊥ f
There are only two non-trivial cases of composition to check which do not immediately follow from functoriality of Lift.
* -> # -> *
# -> * -> #
For the former, we compute that maybe ⊥ g . 𝜂 . f = g.f which is desired and for the latter we need 𝜂 . g . maybe ⊥ f = Lift (g.f) but that is not the case, since 𝜂 never produces a ⊥ but Lift (g.f) preserves ⊥. So either I have missed an alternative way to make Comp a functor or your semantics ⟦_⟧ is not functorial.