Can we prove that a time machine is impossible using the Halting Problem?
When we say that no Turing machine decides the halting problem, we mean just that – the mathematical model of a Turing machine has a certain limitation, not being able to compute the halting problem. Turing machines are supposed to model what is computable in the physical world, but this is not a mathematical fact. If the physical world turns out to be richer, all results on Turing machines will still hold, it's just that their real-world interpretation will be weaker.

What you show in your question is that if time travel is possible, then Turing machines are not a good model for the notion of computation. The fact that Turing machines cannot solve the halting problem has absolutely no bearing on the reality of time travel.

Moreover, you misrepresent the Turing machine model. A Turing machine has no access to the real world, so it cannot halt if it rains on a specific day. All it has access to is its (potentially infinite) tape. In fact, Turing machines cannot be realized in full generality in the real world (since their tape is potentially infinite) – they are abstract machines which are supposed to model some features of reality.