I'm presuming this is happening at some non-trivial relative velocity on the grounds that it's hardly a "big bang" if it's actually in the same velocity vector as the location in which it occurs, and doesn't have any new material coming out.

You're kind of overextrapolating the "we know nothing about pre-Big Bang conditions" based on the English phrasing of a more mathematical idea. We can eliminate some things, like as we are discussing, the possibility that the Big Bang occurred in an already-existing universe in some way that it could create a visibly older star showing up a mere 100-200 light years from us. I can't eliminate the possibility that there's some "collision" happening out of our past light cone, or that a Big Bang occurred and that taking into account metrics of space this time, that it simply "blew" all the old universe out of our past light cone as our universe inflated. But I can eliminate the idea of some older universe being something we could co-exist in local space with. It would produce visibly obvious results that we do not see, a fairly violent cataclysm that would have results that we could see. As we parse the Cosmic Microwave Background, we're parsing twitches and fluctuations far, far smaller than such an event would have created. This would be writ large on the sky, not something that could sneakily inject a star into our galaxy that we only notice after hundreds of years of astronomical observations.

We can't make very many positive assertions about pre-Big-Bang conditions, but we can make a few negative ones like this.

I'd also point out it's not an intrinsically crazy idea. Some variants of String Theory hypothesize situations in which it would be possible for a "Big Bang" to occur in an existing universe like ours, due to membrane collisions. It could well have happened elsewhere. It just doesn't seem to have happened in our past lightcone.