Hi
"Panasonic are telling me my DVD player won't play stuff recorded onto rewritable media, so I'm making some progress."
You are correct...The following is from the s27 user manual..
Discs that cannot be played
DVD-Audio, DVD-ROM, CD-ROM, CDV, CD-G, DVDiR, iRW, DVD-RW,
SACD, Divx Video Discs and Photo CD, DVD-RAM that cannot be removed from
their cartridge, 2.6-GB and 5.2-GB DVD-RAM, and “Chaoji VCD” available on
the market including CVD, DVCD and SVCD that do not conform to IEC62107.
There are also some limitations using CD-R and CD-RW discs as well
And it seems that DVD-R discs might only play if their contents are finalised on Panasonic equiment .
Looks like a new DVD player might be in order. 
Nat