You can simply discard the right hand rotational constraint and project the arm onto the right-hand side of the target previous to each FABRIK iteration.

The solution in the previous reply is not a fix to this (I think it even shows in the example video).

The algorithm doesn't really malfunction or "get stuck", as the original questioner correctly noted, since it tries to right-hand rotate to the target when right-hand rotation is not allowed. Applying left-hand rotation to reach a right-hand target is not doable with FABRIK, except for the above work-around.