Flat Towing a Convertible Automatic

well, let's see...
I can CONVERT my hardtop Jeep to a soft top, and then I can remove the side and rear panels and roll them up, and then I can fold the front of the top back, or fold the top down completely, CONVERTING the Jeep to a fully open vehicle.
Then, in just a few minutes, I can have the Jeep fully enclosed again.
Seems to me that meets the definition of "convertible".
Yes, it is more work and time than just pushing the button on the PT Cruiser, but the Rubicon WILL convert to a fully open vehicle!

Wrangler's with a soft top aren't convertibles, but they sure will meet all the criteria and for all terrain, as well.

PT Cruiser. Towable 4-down with a lube pump. I tow mine on a dolly, no problems.
Chrysler Sebring convertible may also be towable with a lube pump. I don't know for sure, so you need to check with Remco.
ANY front wheel drive convertible is dolly towable.
A dolly isn't as bad as some seem to think. But, if you are dead set against using one, a lube pump may be your only option.
good luck.

Jeep

It would be useful to state what sort of Vehicle You are considering!.

We tow a Geo Tracker that is a Convertible. Although Ours is a Standard they are available with Automatic Transmission as well as Four Wheel Drive! All are towable four wheels down.