Beyma makes quality drivers and the 18pwb1000fe is a typical reflex driver aimed at 30 Hz performance. It does like a big cabinet though, so if space and size aren't an issue, it's a good choice even for the living room
You will have to simulate your box to determine the actual sensitivity in the bass region, but taking it as a guess ~90dB/1W/1m your Bryston is rated for 300W which gives you 114.7 dB/1m not taking into account any room gain effects. This should be more than enough for music use in a domestic environment.
Pnotus: If you mean Vas by bix size, that is not exactly how that works. It is just a parameter, not requirement. All these are fine speakers, but it is hard to recommend one without particular goals set. Kipman responded well regarding expected output. If you just want some fun and more bass, you can go with any of these speakers. These will do the job.
Pnotus: It can show flatter response in the larger box volume, but it particularly does not need larger box. What I see more important in this case is displacement volume, sturdier cone and power handling/input capabilities. For these reasons, I would incline to the Beyma as better choice.
I like to look at the Qts x Vas product (or Qes x Vas) nowadays, the bigger the number the larger the preferred cabinet dimensions are (though by no means necessary). It's an arbitrary number that's 120 for the Beyma and 133 for the Faital Pro.
The Faital does have a lower Mmd, mainly because of the smaller voice coil (3" instead of 4") and the smaller Xmax. Check out the winding depth and gap depth to get the full story of the excursion capabilities of these drivers.
In my experience, if you're going to use up till say 300 - 500 W of amplifier power, you might be better off picking the Faital as the Beyma isn't going to show any benefits at that power. However if you're going for 500 - 1000 W or more and/ or some ridiculously low tuned cabinet the Beyma is going to shine and the Faital is going to get stressed out.
I simmed both. It seems that if you push these and EQ to the mentioned power of 300W and excursion capabilities, the Beyma driver provides flatter FR and 2dB more on average between 10-35Hz, is even at about 50Hz with Faital and then loses 1.5dB at around 80Hz where it should be cut. So I still see the benefits with Beyma.