If I double (halve) the mass in the visible universe, will my inertia double (halve)? I think it should, at least as soon ''I have received the news of this doubling''. But what does the General Relativity have to say about the matter? Effectively I mean...I'm aware that in GR mass, officially, is not there to ''resist force'' but to bend space-time.
There is nothing in General Relativity that links your inertia to the matter/energy density of the universe.
I suspect you are thinking of Mach's principle, or at least some variant of it, but GR does not embody Mach's principle. I'm not aware of any widely accepted theoretical approach that is based upon Mach's principle.
Your question is not well posed. In particular you have said what you mean by mass or by inertia.
The word mass is nowadays exclusively used by practicing physicists talking to each other to defer to rest mass. And gravity isn't caused by rest mass. Current gravity is caused by last gravity. Which is the only correct way to understand gravitational waves but is too glib to explain gravitational sources.
So lets talk about sources. The first thing is to be more detailed about the case of zero sources. For that just note that vacuum spacetimes can naturally be curved without any sources around. And they can curve in different ways or types. Just like electromagnetic waves of different wavelengths and amplitudes can travel through spacetime.
Gravitational sources allow two different types of naturally curved spacetimes to line up where the source is. And rest mass is not a gravitational sources. The sources are: energy, momentum, pressure and stress.
Yes, when you have rest mass you also have energy and sometimes also have momentum and/or pressure and/or stress. But it's the energy, momentum, pressure, and stress that are the sources, not the mass.
And for many everyday situations the biggest source is the energy and the energy is very close (numerically close) to being proportional to the rest mass.
But here is another issue: the rest mass of a system is not the sum of the rest masses of the parts.
And it's worse. There is no global sense of simultaneity so there is no obvious way to even add up rest masses or talk about a system at a fixed moment.
A linearized theory is going to obscure the lack of a simultaneity. And you didn't define inertia anyway. And it's all a bit silly, you can predict how things evolve and so you can make the testable predictions. So what else do you want?