I also suffer from this bug (if it is in fact one). I am also upset about Apple's apparent indifference towards the problem. I think it wouldn't even need a new iOS update. Fixing and replacing the faulty Podcasts app would solve the problem.
That said, I am glad to have found a workaround, which I would like to share here. It's a bit awkward and it will not work for everyone, but at least it's free. Here it comes:
This description is for a Windows system. Mac users may have to adapt the steps.
You will need a metadata editor. I am using MP3tag for Windows.
1. Download the video podcasts to iTunes. My solution will not work when streaming video podcasts.
2. Locate the podcast files on your hard disk. On my system, they reside in subfolders of %USERPROFILE%\Music\iTunes\iTunes Music\Podcasts
3. Copy the files to a convenient location, e.g. %USERPROFILE%\Videos
4. Edit each copied file with MP3tag. Change the Genre tag from Podcast to <blank>. Go to the Extended Tags. Remove every tag whose name starts with PODCAST. Save your changes.
The above changes make iTunes think that the file is a regular video and not a podcast. Maybe a subset of the changes in point 4. will suffice, but I didn't want to experiment too long.
5. Connect your iPad to the computer.
6. In iTunes, open the Movies section of your iPad on the sidebar.
7. Drag and drop the edited podcast files from Windows Explorer to iTunes. iTunes will copy the files to the iPad as movies instead of podcasts.
You can now watch the podcasts in the Videos app, which, of course, supports full screen viewing.
I hope this helps.
Cheers,
Uli