As for .HT_Name, I'm assuming it's the same issue as .CurrentHandler.
It presumably partially auto corrects to fit the save's OT.
If the mon's OT data is the same as the save, then perhaps it can be changed.
If the mon's OT data isn't the same as the save, .HT_Name has to bear the save's name? That's just my guess.
So to modify CurrentHandler, one of the details, either OT or HT, needs to match the save's OT, that's what I think.
I just did a test where I made the HT follow the save's OT, and I was able to edit CurrentHandler (it was on the OT section of the mon, and I forced it to the HT section)
[all edits mentioned are batch edits]
As for Hyper Train Flags suggest, got no clue really.