Whether there is a "true value" of $\mathbf{BB}(745)$ cannot definitively be settled by real-world experiment. Imagine that we run the $745$-state Turing machines and in parallel search for proofs that they halt. We'll end up with some that halt (getting us a candidate for $\mathbf{BB}(745)$), some that provably never halt (which we can ignore), and at least one for which we can never be certain either way.
If we're very lucky, the only leftover Turing machine is the that only halts if ZFC is inconsistent. In this case, we can say, "assuming ZFC is consistent, $\mathbf{BB}(745)$ is this value, otherwise we're not sure yet".
In practice, there will almost certainly be lots of Turing machines we don't understand, no matter how long we keep going for.
At the very least, there cannot be "competing true values" of $\mathbf{BB}(745)$. If I claim that $\mathbf{BB}(745) = x$ and my friend claims that $\mathbf{BB}(745) = y > x$, then we can see which of us is wrong by running all $745$-state Turing machines for $y$ steps. If there is one that halts after $y$ steps, then I am wrong (but my friend might not be right). If not, then my friend is wrong (but I might not be right).
A non-standard natural number is in some sense, a figment of the imagination of a consistent but $\omega$-inconsistent set of axioms. Suppose I have a function $f \colon \mathbb N \to \mathbb N$ and a set of axioms in which, for every $k \in \mathbb N$, there exists a proof that $f(745) \ne k$. (Note that there is not, necessarily, a proof that for every $k \in \mathbb N$, $f(745) \ne k$.)
Then this set of axioms is broken and bad and models of it are modeling nonsense. But if we're working in this set of axioms, we can work with the value $f(745)$ as though it were a natural number, use it in proofs, and so on. In a model of these axioms, there will be an object we call $f(745)$, though that object is not something that is reached by counting up from $0$.
Is $\mathrm{Tree}(3)$ standard? Well, we have a proof in ZFC that $\mathrm{Tree}(3)$ exists, so ZFC "thinks" it's standard. Assuming the $\omega$-consistency of ZFC, that settles the matter.
But in some sense, working with $\mathrm{Tree}(3)$ is a lot like working with non-standard natural numbers. If you spend your whole life counting (or even using "normal" arithmetic operations like multiplication and exponents and addition) you will never get a value that's not provably smaller than $\mathrm{Tree}(3)$, and that is the same experience we'd have if ZFC were $\omega$-inconsistent and $\mathrm{Tree}(3)$ were a nonstandard natural number.
As for Turing machines that halt in a non-standard number of steps - well, in practice, they don't halt, because if you run the Turing machine for $k$ steps and it halts, the description of your computation is a proof that it halts in $k$ steps that any consistent model of arithmetic should accept. But, well, what can you do with a Turing machine that hasn't halted yet?
You can try to prove something about whether it halts or not. And the $\omega$-inconsistent set of axioms you've chosen will supply such a proof - it will prove that it halts! It will prove that it halts without giving you a way to compute how many steps it halts in, and in fact, for every $k$ you name, there will be a proof that the Turing machine halts in more than $k$ steps.
Again, because we don't know that the axiomatic systems we work with are $\omega$-consistent, this will feel a lot like looking at a Turing machine that runs for $\mathrm{Tree}(3)$ steps. ZFC claims that this Turing machine halts, but in practice, it will not halt in any real-world computation, because any real-world computation will not run for long enough. For every $k$ you name that you can count to, there will be a proof that the Turing machine halts in more than $k$ steps.
