The strings Jamerson played on are best represented nowadays by the Labella 1954s. They're bridge cables.
Always use your mid frequencies to your advantage. They sound ugly when you're alone, compared to a nice scooped "bedroom tone", but they fill a mix nicely—and if you really push them, you can start getting into territory like Lemmy (albeit, he used rounds, obviously). Bass 0, Mids Full, Treble 0, Presence Full.
Different makers' flats are definitely different, though. Labellas, whether the DTFs or LTFs, are very bassy and thumpy. GHS precision flats are similar, as are a lot of others. TIs are pretty flexible and maintain some brightness. Rotosound 77s are very zingy at first, and if you keep them really low and play hard you get a lot of Steve Harris's clanky sound, but they're still an old school flat, especially as they mellow with age—just less thumpy, more round and actually-punchy, in my experience. D'addario Chromes and EB Cobalt Flats are much more bright and modern, Cobalts especially, because they don't really lose it.
They'll also have different levels of polish. Labellas are very highly polished, with 1954s to my memory looking almost like solid wire. DR Legends are similar. Rotosound 77s and EB Cobalt Flats, meanwhile, you can actually see the wraps a lot more obviously, and that contributes to the grippy feeling they have—not as rough as rounds, but there's definitely something there.
Also, definitely experiment with some overdrive and distortion on the flats. Even just a little bit of break-up with flats is a fantastic sound, and it's what you hear on a ton of rock bass parts in the early 60s.