Family Man and many others use a J, but if I recall correctly, he also solos the neck pickup, so it kind of functions in the same range.
Culture Club's "Do You Really Want to Hurt Me" is an excellent example of something different. Pretty sure it's a Stingray and it still sounds really great for reggae, even if it's not a typical sound.
Also a nice hack is that One Control Bass Preamp. It can make any bass super thick for reggae without the boom of just EQ.
Great bass track on that song. Man the Brit bassists in the 80's...they really got it. Another couple examples of IMO great "reggae-ish" bass tracks on pop songs--"Watching the Detectives" by Elvis Costello (with the great Andy Bodnar on bass) and "The Tide is High" by Blondie with the also very great Nigel Harrison. Neither song is real reggae, nor is either bass tone (they just sound like good, serviceable Pbass sounds to me--I could be wrong), but the spirit of it is there. Bodnar does it with a lot of really cool variations and movement, while Harrison plays a deceptively simple calypso-inspired feel with very subtle variations--so good.
If you want to use flats that’s fine, but there are lots of reggae bassists who use rounds.