I haven't used the newest version of the BFG AT but I have an extreme dislike for the previous generation of BFG AT and think anything is going to be better than those (yes, I have owned many sets and hated every one and kept returning to them because the internet said they are great).
Anyway, I haven't used the Falken's but have several friends that have and really like them. They will be on my short list of options for the expedition. Other than deeper mud the Falken should do almost everything you would want out of a tire. My friends have even gotten a lot of miles out of them on their trucks. We don't get a lot of mud in Utah, so mud usually isn't a big consideration when picking AT style tires. Desert, rocks, and some winter roads are what we see most.
I am not quite ready to put the new BFG on my list yet. The new Goodyear wrangler ultraterrain AT's look nice but I doubt they have a 22" version of that one either. I suspect there are not a lot of 22" options out there. The Toyo AT2 is a nice tire and is available in some 22" sizes. I had those recently on my jeep commander and really liked them, great all around tire.