Is there a way to experimentally determine the true mass of C-12 isotope which is defined exactly as 12 amu? As most you know the measurement of atomic masses has > 200 year old history and atomic masses were deduced on the basis of chemical reactions by meticulous chemists until the 1940s. Later, mass spectrometer became the tool for determining atomic masses. In each case, scientists assigned one of the elements an arbitrary number and calculated the masses relative to that. For example, early chemists like Dalton, set H as exactly 1. Later, O-16 was set to 16 exactly. By the 1960s, C-12 was chosen as the reference and its atomic mass set to 12.  Today, all atomic masses in the periodic table are relative atomic masses with C-12 as a reference.
What puzzles me is that if all atomic masses are relative to one particular element, what is the actual mass of an element? All these definitions were chosen by chemists without any theoretical justification in the past. I mean O-16 was set to 16 when the concept of atomic structure was not even clear.
With the experimental availability of Kibble balances and perfect silicon spheres, would it be possible to get rid of relative atomic masses in future and find the true (absolute) atomic masses of elements? 
P.S. Edit 
The mass of carbon or oxygen atoms originates from a circular definition (as posted in the answer). All these masses are calculated on the basis of assuming that 1 mole of C-12 atoms weighs 12 g exactly. There is  a long long history behind it. I didn't want to repeat all here. I am not worried about kg or anything else. All I am saying is that would new approaches like the Kibble balance allow us to get rid of relative atomic masses?  
 A: A quick search in google will give you the mass of a carbon atom as $1.9944235\cdot10^{-26\ }kg$ and that of oxygen atom as  $2.6566962\cdot10^{-26\ }kg$.
These are absolute masses of the elements in SI units. But, if you think about it, is it any intuitive to use these masses for calculations?
The current atomic mass unit is extremely convenient, and (though I don't have much of an idea about it) I think that physical chemistry has to do a lot with concepts of moles and molar masses, which are rendered far easier and logical by using the relative masses.
By the way, your idea of a theoretical justification is faulty. We have as much theoretical justification for using kilograms as the SI unit as we have for using atomic mass. A kilogram was just the mass of a platinum-iridium cylinder (IPK), and is now defined using Dimensional analysis (if I remember right) after fixing values of second (s), metre (m) and Planck's constant. 
In short, almost every one of our measurement is relative, and we use the one most convenient for a particular situation    
A: The integer numbers come from the experimental observation that atoms are composed out of positively charged nuclei with negative electrons bound around them, and the number of electrons neutralizes the number of protons in the nucleus. So each atom is accorded an Atomic number, which is an integer and characterizes it in the table.
What you call atomic mass units the wikipedia has as Dalton units. In honor of Dalton who first recorded the periodic table which is now ordered according to the atomic number of the elements (i.e. the number of charges ).

The dalton or unified atomic mass unit (symbols: Da or u) is a unit of mass widely used in physics and chemistry. It is defined as 1/12 of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest

It is useful in order for people doing calculations not to have to carry kilograms or other mass units in the microcosm of atoms and molecules.
