I'm not too convinced that this has much of a long-term effect. How much total notoriety do you get from the BP quest? 10000? 15000? 20000? That's less than one level's worth once you hit the high 20s.
But earlier on, it really makes a difference. How much rep you get depends on both your overall notoriety and the current weapon level vs the level of the mob (I could be wrong, but I think it works that way). It's possible to get to level 10ish with mostly quest exp.
So you're level 10, but have level 3 in weapons. And suddenly the mobs that you can kill with your current weapon skill levels don't give you much exp. So you have to grind a bit. That's no fun. Plus, when you level weapons, you get more hit points and more voodoo. Not as much as a full notoriety level, and the amount depends on the weapon, but it adds up. That 40 HP you get from going to level 3 to level 8 on cutlass can make the difference between killing that mob for even more XP, and a trip to jail.
Here's what I'd recommend:
1. Don't do the BP quest until you're in your late 20s, and they've fixed the boss battle. Really, until they do that, there's no point.
2. Only do the weapon and teleport quests, or quests that give money. There are several of those on Port Royal, and I hit them as often as I can. In particular, Bartholomew Watkins (I think that's his name). He hides behind some barrels, and will give you 105 gold for every boat trip where you sink something. It adds up.
3. Try to keep your weapons at roughly the same level. When you level one, switch to another to level it up too. When you get a new weapon, concentrate on it to get it to parity with the others.
Since I finished with the BP quest, I've been concentrating on my weapons. I've got them all to 15/16, except staff (8) and grenades (10). Just got the staff, and spent all day yesterday getting them and the grenades up (8 staff levels, 4 grenades). The HP I got from those made a big difference in my survivability.
I made a third character for the screenshot contest (Vote for "They're always after me Lucky Charms!"
here!) and decided to keep playing him. I skipped the BP quest with him, and have noticed that the weapon/teleport quests are a little easier, as I have more HP than I did with the last character, plus he's better with his weapons.