If I don't get satisfaction from HP on this, is it likely that flashing an update will improve my burning of movies?
I have no idea if things improve with other firmware. I personally own a TS-L632H (Dell OEM), and I found no significant difference in burning quality with Dell OEM and Samsung retail firmware. This might not be necessarily true with your HP firmware. You may try at your own risk. To judge about the quality, you can do some scans (see FAQ) and compare then.
If you want to burn often, you may consider buying an external Half-Height drive. These are more reliable.
I've already invested in -R and +R TDK and Verbatim discs as well as trying different burning speeds.
TDK is a great name, but unfortunately they are not different from random supermarket brands Stick with Verbatim.