I'm a lefty, playing lefty, who has invested a lot of hours, on multiple occasions, in trying to switch over to playing righty. (both on bass and guitar)
Never worked for me.
What I have found is that while my left hand is great at fretting, my right hand cannot function rhythmically at a satisfactory level, period.
I have heard this same observation from many lefty-playing-lefty players, as to the dominant hand being needed to pluck/pick/slap at sufficient tempo and rhythmic "tightness".
Heck, even Steve Morse, probably one of the most famous lefty-playing-righty guitarists (and an absolute virtuoso at his instrument) remarked on having the same problem with his right hand!
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If Steve Morse has this issue, what hope for us mere mortals?
All that being said, the only right answer is the one that feels right to you.
Left/right handedness isn't a black/white thing. It's a continuum. Some lefties are really lefty (raises hand). Some have a greater degree of ambidexterity.
From what you've posted of your experience so far, I suspect that playing lefty might be the better option for you. If holding the bass left-handed feels significantly more "correct" for you than the other way around, this is a good indicator.
The question isn't "can you, as a lefty, be good at playing righty". I'm convinced the great majority of lefties can be. The real question is "what compromises will you make in your playing ability if, as a lefty, you play righty." For some, the answer is "minimal". For others (me), the answer is "significant".
As far as string order goes, there are several very good bassists out there playing lefty who have the strings "strung righty" with the biggest string closest to the floor. Nothing wrong with that, and you can do some really cool stuff with that configuration, but realize that there are practically no books, videos, or other teaching aids that accommodate that configuration, so you will have to "remap" everything in your mind.
One other thing that playing righty-upside-down gives you is access to many more basses, if you don't mind the controls being by your elbow.
On the other hand (ha!), lefty basses are much more common than they once were, and much easier to find.
Regardless, don't let equipment availability be the determining factor in your choice.
I hope this helps.