I used the Sharepoint explorer and saw that the property of the view  is hidden. I tried to change the hidden property of the view but the Sharepoint Explorer from Navigo does not allow me to change the hidden setting.
I'm also very interested in a solution...

Don't try to replace  the LVWP webpart. Just add the DVWP to the page
and add a CEWP part  to 'turn off' the LVWP through javascript. This
way, the backend side of things thinks everything is normal; it's just
the UI that is fancy.

It's messy and I wouldn't like to support it but it will work.

You can also use the ?toolpaneview=2 querystring on the view  page, and
then Close the LVWP, rather than hiding it or un-checking the Display on
Page property.

Both closing the Web Part and/or making it not visible on the page cause
the list  view to drop out of the left  nav. Are you suggesting something
that doesn't cause this anomaly?

This is what I did on the page where I wanted to use a DVWP.

I went to the All Items page, and appended "?toolpaneview=2" to the URL.
The page showed up with the web  Part Zones, and the main view  as the
only web part. When I clicked on the X in the corner of the list  view
web part  (which if I understand correctly, is "Closing" it), the LVWP
disappears, but the View in the Views list does not. Then I switched
from Browse mode to import mode, located my .dwp file for the DataView
Web Part, and dragged it into the zone.

I have seen where it makes the view disappear from the Views list, but
as far as I can tell, that only happens if I uncheck the Visible on Page
property.

Is that the same thing you are doing? If so, I can't explain why it
disappears for you and not for me.

I'll give it a shot....................