Would inertia grow differently if we switch the reference frame?
LHC accelerator makes protons x7000 heavier at record speeds and I just wonder if we could change the reference frame (keeping velocity), would its mass be different? (eg. in vacuum, far from any gravitational field). If velocity is relative, we can say that protons are in rest and any other frames are moving at that velocity in opposite direction...
Your main problem here is that you are mixing ideas from two different ways of formalizing relativity.
If you are using the (old fashioned, unnecessary, and easily misapplied) notion of "relativistic mass" then mass is not invariant.
If you are using the modern nomenclature then 'LHC accelerator makes protons x7000 heavier' is incorrect, in that framework the accelrator gives them an energy 7000 times their mass energy, but their mass remains exactly the same.
Now, you add 'inertia' to your title, but I'd like to talk about why that doesn't actually make relativistic mass a good idea.
Let's say that you have a mass moving a speed $v$ with respect to you and you apply a force $F$ to it. From Newton's second law you would expect that the ratio of the force to how fast the velocity changes to be the mass.
If you apply the force perpendicularly to the direction of current direction you'll get $$ \left(\frac{F}{a}\right)_\text{transverse} = \gamma m \;,$$ where $m$ is the mass you measured for the object at rest and $\gamma = [1 - (v/c)]^{-1/2}$ is the Lorentz factor. This is the usual "relativistic mass"
But if you apply the force along the same direction as the current motion you $$ \left(\frac{F}{a}\right)_\text{longitudinal} = \gamma^3 m \;.$$
So the relativistic mass isn't the inertia in any general way.
In the modern parlance, this difference is understood to be a consequence of the way velocity composition works, because the mass remains the same.
