What prevents BFS from becoming unstable when pitching from horizontal to vertical?
The answer here is to review the history of how Space X "does it" with the Falcon 9.  To solve aerodynamicly, they extend speed brakes from TOP of booster to get it to fall tail first.  CG buffs take note, this is how it works in gravity and atmosphere. Just like a parachute or a hang glider.

Very close observation of the video shows the lower "tail fins" folding inwards, raising the aerodynamic pressure point (in this case drag) above the CG.  The rocket now behaves as an aircraft with its CG way too far back(relative to Clift), pitching up and falling "backwards" towards the ground. Rocket thrust then slows the descent for landing.

The rocket is rotating during this maneuver to its new weight down and drag/lift up configuration, with predictable increase in velocity due to lower net drag towards the direction of "flight".

It may be helpful to draw the "lift" vertical and horizontal components here to understand the forces.
The maneuver is not unlike a 1/4 loop.  It is controlled, because CG is below the Center of drag.

In response to edit regarding yaw axis, great observation!  This is a technique considered for airliners recovering from deep stall!!!  Rock back and forth to bring V stab into play to yaw out of deep stall.

But with the rocket, the gimballed motor will stop this, and before, rudder will now not be used to break the stall, but to preserve it!  This is an aerobatic maneuver, pure and simple.

What Space X does so well is transition the "deep stall" into a controlled landing.

The concern about the V stab is justified.  The Falcon 9 speed brakes are more fool proof.  Let's hope they do not "engineer themselves into a corner" with a potentially flawed design.  

POST EDIT - RESPONSE TO COMMENTS

We need to keep in mind here that the animation is what may be wrong, not the actual flight plan. @qq jkztd correctly pointed out that, if the BFR was falling vertically, with no horizontal motion, the pitch up would cause horizontal motion towards the tail (beginning to glide backwards). Although the pitch up would not induce yaw, the backwards gliding BFR WOULD be unstable in yaw. Folding in horizontal fins would make it stable in pitch. A better solution may be to initiate pitch up with some forward motion, or to simply pitch up by adding more drag to "top" with speed brakes as the Falcon 9 does, or igniting gimballed rocket motor sooner.  However, sometimes, simpler is better,