Can we determine an absolute frame of reference taking into account general relativity?

The problem is that to determine the distribution of mass in the universe you need to choose a coordinate system that you're going to be using for measuring the positions of all those masses. The trouble is that you are free to choose whatever coordinate system you want to make this measurement. There is no absolute coordinate system for measuring the mass distribution. Your choice of coordinate system will determine how much of any acceleration you measure is inertial and how much is gravitational.

The four-acceleration is given by:

$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha{}_{\mu\nu}U^\mu U^\nu $$

and speaking rather loosely the first term on the right is the inertial acceleration and the second term is the gravitational acceleration. The problem is that while the four-acceleration is a tensor the two terms on the right are not. It is always possible to choose a coordinate system that makes the inertial acceleration zero - in fact this is simply the rest frame of the accelerating object. Likewise it's always possible to choose coordinates that make the Christoffel symbols, $\Gamma^\alpha{}_{\mu\nu}$, equal to zero - these are the normal coordinates.

This is the equivalence principle in action. While the four-acceleration is a tensor, and therefore a coordinate independent object, the two terms on the right can be interchanged by a choice of coordinates making the acceleration look purely inertial, purely gravitational, or some combination of the two just by changing coordinates.

Since there is no absolute coordinate system for measuring the mass distribution there is no absolute coordinate system for measuring the inertial acceleration. The two types of acceleration are fundamentally indistinguishable.

I like John Rennie's answer, but I'd like to add something. From the point of view of general relativity alone, and equations of motion, his answer is complete. From the point of view of large-scale cosmology, there is something to add. It turns out that the distribution of matter in the universe is rather simple on the largest scales, in that the evidence is that it is homogeneous and isotropic (as I say, on the largest scales). It follows that one can use this matter distribution to set up a most natural frame of reference or coordinate system. This is the reference frame in which nothing is moving on average at the largest scales. It is called comoving coordinates in cosmology. One can discover one's acceleration relative to this coordinate system.

There are two underlying issues here, which are both conceptual, not mathematical.

(1) The question is based on the newtonian world-view, which we find empirically does not describe our universe.

there should be a determined distribution of mass in the universe [...]

The hidden assumption here is that we have some newtonian universal, absolute time parameter, so we take a snapshot of the universe at this time, and find the mass distribution. There is no such universal time parameter. There is no well-defined notion of simultaneity on the cosmic scale.

[...] and assuming it is known, its effects should be able to be subtracted to deduce 'absolute' acceleration.

This would involve adding a bunch of force vectors, but that requires transporting all the force vectors to one place. Parallel transport is path-dependent.

(2) The question proposes using global information in order to determine what is and what is not an inertial frame, and then asks, isn't this a prescription for finding an absolute frame of reference? This is mixing up the roles of velocity and acceleration.

GR, like SR and Galilean relativity, does have a distinction between inertial and noninertial frames. However, we don't need (and, in the case of GR, can't get) global information in order to do this. Local experiments can tell us what is and what is not inertial motion. If we release a test particle, its motion is inertial. Any motion that is at constant velocity relative to the test particle is inertial.