I have to be careful what I say here, lest I get cornered in by my own spontaneous thinking on this matter.

As Batchelor says Buddha was born into a Vedic culture that accepted reincarnation as an accepted given. But is it a given now If Buddha believed the earth was flat (quite likely) and upon his subsequent enlightenment/awakening did he know that the earth was really a sphere? I doubt it's even possible. He could have gone on seriously thinking the earth was as flat as a pool table (figuratively speaking) long after his greater spiritual realizations of other more important things as they were pondered, and without impinging the magnitude of his enlightened being one iota. Did he know about atomic structures and quantum physics? I doubt any of that was ever crossing his mind even once. We don't really know about reincarnation empirically speaking, and never will.

Did Buddha talk about rebirth? Yeah he did. But did he mean reincarnation, or just simply rebirth of a brand new person within. I think the later. He also talked about karma as it was the standing metaphysics of his time. Did he mean karma (as cause and effect) in relationship to one's place in the greater scheme of things regarding reincarnated states of future lives (i.e. metaphysically) or did he mean karma (as cause and effect) in relationship to being our own authors of bad luck, demise, ill will and so on, and/or better fortunes to come by being positive in this right now existing life?

When you read Batchelor's writings does he insist that rebirth is only about reincarnation alone? Does he insist that karma as it's tied to it, is only about reincarnation, and not a more general sort of rebirth of the person living, and still breathing?

If he is connecting the two as inseparable, is he in reference to strictly reincarnation? Does he widen his thesis ever, regarding karma as a cause-and-effect and a tool or means to connect to the rebirth of a person into a greater being, while still alive and breathing here and now?

I think no one can answer these things with certainty as far as what Buddha taught. Not even Batchelor! We can only surmise things as they may have been meant by Buddha.

I will have one perspective the rings out for me, you might have still another, Batchelor certainly has his beliefs, and so on; for every person under the sun.

I respect Batchelor and what he writes about Buddhist theosophy/philosophy without holding deep dogmatic beliefs. But I doubt that everything he says will wash with me 100%. That's no reason to reject everything he teaches, simply because I see he might be all wet on one or two or three issues. First I don't pretend to know everything; I doubt Buddha did either, he was just head and shoulders above the crowd in his time (way over their heads). Second if Batchelor says the moon is made of green cheese will I believe that? No it's not even possible. Is Batchelor confused about some things? Maybe.. even quite probably. But does he have something to say about Buddhism as a non-dogmatic endeavor? I think he does. And he does so very clearly, but only if people choose to listen.