You are the one being insufficiently precise here. There would only be fundamental obstacles if there was an additional claim "allows all correct programs".

The Halting problem states only that is no computable function that takes another P program as input and always terminates with a correct answer of whether P halts. It's certainly possible to write a program that always terminates with an answer of either HALTS or UNKNOWN, and only says HALT when that's true, it's just that it will also return UNKNOWN for some (or all) programs that do actually halt.