Symmetry in Hawking radiation?
This assumes that when a pair particle-antiparticle is created, a bias towards antiparticles being captured and particles escaping must exist.


No such assumption or bias exists.

As far as we know the universe treats anti-particles just like particles with different properties (e.g. charge is opposite compared to the corresponding particle).


  Otherwise, the mass of the black hole might fluctuate, but in average for big numbers of pairs, equal percentage of particles and antiparticles would be captured/escape.


You seem to be implying that anti-matter has negative mass.  All particles (including anti-matter) have either zero mass (like photons) or positive mass (like electrons).

So regardless of whether the anti-particle or particle escapes, the black hole looses mass (more precisely it looses energy).


  So, is there really a bias needed for the black hole to loose mass, or is it something wrong about my understanding of Hawking radiation?


Your understanding of the mass of anti-particles is likely the problem.  They have positive mass, just like particles, because they are particles.