Ways to get badges starting now:
Run Karazhan once a week: Karazhan is probably the grandmaster of badge sources these days, to the point that some people have dubbed it Badgazhan. With the Huntsman giving 1 badge, Prince Malchezaar 3 badges, and all the other bosses 2 badges, you can get 22 badges every week from Karazhan alone. That may seem like a tall order sometimes, but it's gotten to the point in my guild where we put together pickup Karazhan runs that can clear the whole instance in 3-4 hours, and that's often with at least half the raid in no better than Tier 4 gear. If you're really good (and have 10 accounts handy), you can just 2-man Karazhan. 10 weeks at 22 badges a clear is 220 badges. That's enough to buy one of the weapons (or a set for you dual wielders) and have a good chunk leftover towards a ring or a piece of armor.
Run heroic dungeons 2-3 times a week: You could always run them every day too, but let's just assume you have other stuff to do, whether it be a real life to run, or just dailies or trade skilling or 25-man runs to take care of in game. You can let the daily quest do the choosing for you, or you can choose to run one of the old badge-heavy standbys such as Underbog or Mechanar. If you do a run two times a week, you'll have somewhere in the neighborhood of 80 badges once our hypothetical 10 weeks are up, depending on where you go and how many of those runs are done with a daily quest. That's good for some gloves, belt, or boots with a bit left over, or a ring with a bit more left over. If your group is decently geared and keeps on their toes, it shouldn't take more than 2 hours per run, so you won't be spending too much time on it.
Run Zul'Aman once or twice a week: This one won't be as easy as the last two options. Zul'Aman is definitely a bit of a gear check, and you'll probably need to find specific classes for a lot of fights and roles more so than Karazhan. That said, a run is still doable for a group of dedicated fighters in a mix of Tier 4, Karazhan, and badge gear, and at a 3-day reset, you can take this on twice a week. A full clear will net you 13 badges, and if you can grab Zul'jin's blood once, that's another 10 badges, so even if you only do one clear a week, that's a potential 140 badges in 10 weeks. Again, this one is a little harder and requires a bit better gear than the previous options, but there's great drops to be had, and it will get you most of the badges you need for a weapon.
More ways to get badges when 2.4 goes live:
Run 25-mans: You may have already heard it right here, but in case you didn't, 25-man bosses will start dropping badges when 2.4 comes out. No need to quit 25-manning so you can run Karazhan again, the badges will come to you. It'll probably be a bit slower than some other options at only 2 badges on each boss, but you'll also be getting that old fashioned dropped loot as well. Then you'll just have to spend your badges to get Vanir's Left Fist of Savagery when Grip of Mannoroth just refuses to drop off the Eredar Twins.
Do Dailies: This ones going to be a bit more streaky, but you may recall hearing that certain Shattered Sun Offensive dailies, both at Shattrath City and at the Isle of Quel'danas itself, will provide a reward of Shattered Sun Supplies. This box will usually contain a high level green item, but it also has a chance of containing a badge. This probably isn't a method you want to specifically rely on to get your badges unless you're super casual, but if you're doing dailies anyway, it might be a good idea to seek out the ones that give this reward and do them. Assuming they keep the chance to drop a badge when 2.4 goes live, it might at least save you an extra heroic run by the time the vendor's unlocked.
So there you have it. It probably does seem daunting enough to have to save up 150 badges for a weapon, not to mention over 400 if you want "one of everything," but don't lose hope. One or more of the above methods may just work for you, and by the time it's time to make ready and get that gear unlocked, you'll be ready to go with enough badges to purchase shinies to your heart's desire. With the stats that stuff has, I think the collection work will be worth it.
Now the one problem I have left is that between my 3 level 70 characters, I probably want somewhere close to 1000 badges worth of gear. Even with all the Karazhan and Heroics I run, I don't think I'll quite make all my gearing goals right away. I may have to choose one of them to focus on initially, maybe two. But that's like choosing my favorite child, you know?