When you got to this line:

The tool looked brand new, but had the earmarks of being a cheap
import.

I knew then you were going to have problems without reading the rest.
I've never had luck getting a cheap import to work for flaring copper
tubing (especially double flares). I made the mistake of buying one of
these pieces of crap once. I had the same problem as you...the tubing
not staying put. I finally found a decent tool at work and that one
worked great.

Mike