I've known several bassists who play the kind of stuff you do, who bought a 6-string, and who ended up never using the C string at all. (And to be fair, they didn't use the G string much either). It ought to seem obvious that to use an extended range bass, you need to play notes extended beyond the standard bass range.
If you often find yourself wishing for some higher notes without jumping up the neck too often, then a 6-string is a good bet. If not, you still may wish to rearrange some of the material you play to use that range, even if it means cooperating with the guitarist!
As for a 5-string, I find I rarely use that low 4th from B up to E, but I use the B string a lot anyway rather than reaching toward the nut on the E string.