Move external HD to a new drive, but have Time Machine think it's the same?
Instead of using the finder to copy your files - use a program like 'carbon copy cloner' or 'super duper'.

This will get you the closest to the two drives being exactly the same.

However, I should warn you that when I did this myself, I could not get Time Machine to treat the new drive as if it were the old drive. It insisted on keeping a backup of it as if it were a new drive. (however, in my case this was the OS internal drive and not an external data drive)

In the end I decided that it was OK that Time Machine didn't know it was the same drive, as I now had two copies of the data (before time machine even backed up) from the original flaky drive, and the new drive.

I just had time machine back up the drive as if it were brand new. I haven't had any regrets about it.

Hope this helps.