Nope. You don't need to "solve the halting problem" and I guess I'll explain why here.
Firstly lets attack the direct claim. The reason you'd reduce to the halting problem is via Rice's theorem. But, Rice only matters if we want to allow exactly all the programs with the desired semantics. In practice what you do is either allow some incorrect programs too (like C++ does, that's what IFNDR is about, now some programs have no defined behaviour at all, but oh well, at least they compiled) or you reject some correct programs too (as Rust does). Now what we're doing is merely difficult rather than provably impossible, and people do difficult things all the time.
This is an important choice (and indeed there's a third option but it's silly, you could do both, rejecting some correct programs while allowing some incorrect ones, the worst of both worlds) but in neither case do we need to solve an impossible problem.
Now, a brief aside on what Rust is actually doing here, because it'll be useful in a moment. Rust's compiler does not need to perform a static bounds check on array index, what Rust's compiler needs to statically check is only that somebody wrote the runtime check. This satisfies the requirement.
But now back to laughing at you. While in Rust it's common to have fallible bounds checks at runtime (only their presence being tested at compile time) in WUFFS it's common for all the checks to be at compile time and so to have your code rejected if the tools can't see why your indexing is always in bounds.
When WUFFS sees you've written arr[k] it considers this as a claim that you've proved 0 <= k < arr.len() if it can't see how you've proved that then your code is wrong, you get an error. The result is that you're going to write a bunch of math when you write software, but the good news is that instead of going unread because nobody reviewing the code bothered reading the maths the machine read your math and it runs a proof checker, so if you were wrong it won't compile.
Edited: Fix off-by-one error