J bass (P bass for reggae is largely a myth, IMO, perpetuated by people that haven't really done it or even really A/B'd the different tones in a J and a P but it seems like you already know that), flat cans help - LaBella Deep Talkin' flats sound pretty huge. So do Seymour Duncan Antiquity II pickups. Any amp will do - you just need headroom so as to be able to push big lows.
Ultimately reggae is a style and not a collection of gear. If your chops are up to it - right hand palm and/or left hand muting - then you should be fine with what you have bass-wise. Rolling the tone off on the bass can help, along with perhaps a higher value tone capacitor in the bass and/or shifting the balance to the neck pickup.
I play reggae just fine with a J bass, roundwound stainless steel strings, both pickups up, maybe rolling off the tone, and usually I leave the amp (Carvin B1500 will give you massive wallop - some tube amps have a nice pillowy low end but I've never needed a tube amp) set the same as I have it for rock or funk. A classic rig would be an SVT and 8x10.
For me it's all in the hand muting.
To copy Sublime tonally though, you will need gear that can put that tone out at the volume you'll need for jamming or some kind of preamp if you're just recording. I play Sublime covers like Bad Fish and What I Got just using the setup I described and it's fine.
