The drive (LG AP70NS50) has always worked without separate power, so I don't think that was the issue.
In any event, the new drive is now working. Not sure what caused it start working, so I can't supply the fix. It started working after I plugged both the new and old drives into USB ports. I had tried re-assigning the drive letter previously, so I don't think that had anything to do with it.
My dock is the one for Surface Pro 4 - the little brick-looking thing. Again, the problem occurred regardless of using ports on dock or the Surface directly, but I'll look into that firmware update you mentioned.
Thanks.