I think Epson only created Media Types in the driver for the Epson papers users would most likely use.
The 3800 can print fine on the Heavyweight; there just isn't a built-in Media Type for it. That's in the same way that the 3800 can print fine on Hahnemuehle Photo Rag, even though there isn't a Media Type for HPR, either.
The only thing is that you'll need a profile for Heavyweight, and not sure where you'll get that from unless you get a custom one made.