In my humble, you may be stuck with using the snake method unless you can either get more ideas or get better equipment to clean the duct (which I don't know of). As for going under the house I've got the feeling that you're refering to going into a nasty crawlspace), you should be able to 'drop' the pipe and disconnect it to clean it out. It's my bet that it's still clogged.
One of the problems with your type of situation is that if there are a lot of 'turns' in the piping, there's more opportunity for the vent to get clogged. Looking ahead, if you can reduce the number of 'turns' in the piping, you'll do yourself a favor. In the meantime, should you end up having to go under the house, put on some long sleeves and long pants and a hat and gloves first. (I'm sure you knew this.) Then, you can get some sheet plastic to 'roll out' as you crawl. BUT, first, take a flashlight to see what's used to 'hold up' the duct and what tools you may need to unhang and dismantle the duct. Have with you replacement parts because more than likely, you may NOT be able to reuse them. I'm gonna bet that once you drop the pipe and disconnect it, you can see what's going on in the disconnected pipe as well as what's inside of the duct that's left hanging.
Have someone outside the under-part of the house to go get tools for you and such. Tools you may need are hacksaw, claw hammer, small pry bar, flashlight, goggles for sure, clean rags, gloves, duct straps (that hold up the pipe), 1" roofing nails (to re-hang the straps), and patience.
Again, if you find out that the ultimate problem is a clogged duct, perhaps reducing the number of turns it takes to get outside will help. Also, the shortest distance to the outside helps too. My best to ya and hope this helps. (Perhaps someone else has a better cleaning method than I.)