I think I've got a way to keep the Descanites happy.
So what we do is this: First, we /extend/ your brain. We build a copy of your brain, but instead of just booting it up, we HOOK you to the copy somehow, so you can the copy are sharing information.
So basically, it's all you, one consciousness, but you've got a redundant copy of all the data. I think you'd agree that this is still you, right?
Then, rather than replace bits, what we do is slowly cut your brain down the middle, so that when trying to access the opposite hemisphere, it's accessing the backup data instead. This is a gradual process. There's a shares consciousness, but it's getting steadily more out of joint. This mirrors a person that has had a partial bisection, in that they have one consciousness but half their brain has access to information the other part doesn't, but there's still continuity.
Then, like a cell splitting, we continue this process. At some point, well beyond severing the last link, it's far more realistic to think of this as two separate consciousness that share a bit of information, but there's no division where that occurs, no break in continuity.
Then, we sever the last link... and now there's two of! There's two of you, with no break in continuity, meaning both of them are you.
Then we simply repeat the process, except this time not bothering to hook the organic part up to anything, and now we've got two of you with a 100% mechanical brain.
I forget why we wanted two of the same person (or at least two people that WERE the same person at some point in the not-so-distant-past, and thus two people with a common ancestor who both have a 100% valid claim to being the same person as the pre-operation fellow), but the transfer to a mechanical consciousness has succeeded!