I assume the stripes are dye just like the rest of the seat. How can I re-dye these seats without ruining the stripes? Surely they'll get some cleaner/prep/dye on them no matter how careful I am.

Has anyone tried to dye just the sides?

Tape it. RO and PC won't do anything to the color, If you tape it, you shouldn't get Prep Agent and Dye over it.