There's a good chance Amazon blacklisted the 'broken' Kindle when they sent a replacement device,
That's not my experience. I was in a similar situation a while ago. Amazon definitely did not blacklist the old Kindle. It was still registered to me by the time I got round to sending it back (which I did voluntarily).
But, of course, that doesn't mean that the same will be true in this case.
I had also done a factory reset after the battery-removal reset to make sure the default firmware loads again, and now the device doesn't show any registration info. My brother in law had registered the replacement K3 on his Amazon account as soon as he had received it. My question is, can I register this Kindle on Amazon under my name?
Just because the device
doesn't show any registration, that doesn't mean it's not registered. You should get your brother-in-law to go the Manage Your Kindle page and check there to see if it's registered. If it is, have him unregister it (from the same page). It should then be OK to for you to register it to yourself.
Even if that doesn't work, I would think the very worst that could happen is that you won't be able to register to old one. I can't see why Amazon would blacklist the new one. But who knows?