I think you may need to process this file yourself - either manually, or by writing a script to automate the extraction.
If you rename the file GMRT.kmz to GMRT.zip you can open it in WinZip or similar, and see its contents.
This includes the file doc.kml, which contains a number of Links - examining the first link shows a URL:
<Link>
<href>http://www.virtualocean.org/cgi-bin/GMRT?wesn=-180,-144,-90,-54</href>
<viewRefreshMode>onRequest</viewRefreshMode>
</Link>
Open the URL http://www.virtualocean.org/cgi-bin/GMRT?wesn=-180,-144,-90,-54 and note that this downloads a new KML file. Open this file in a text editor and see the first ICON image:
http://www.virtualocean.org/data/tiles/geographic/geographic_512/0/0/0_7.jpg
You can download and georeference this image using the coordinates in the KML file:
<LatLonAltBox>
<north> -54 </north>
<south> -90 </south>
<east> 108 </east>
<west> 72 </west>
</LatLonAltBox>
So if you're unable to process this file automatically using tools in QGIS, you could automate the process using the above manual steps.
Best Answer
Use the "Iterate Feature Class" tool and set up your model to look similar to this:
To get the name of the output KMZ to match the name of the input shapefile, you will need to make some changes to your "Layer to KML" tool.
Set it up to look like this, using %Name% as the name of your desired output KML in the Output File line:
Setting the name like this is called inline variable substitution. For more information, take a look at this article.