First of all, dcgeorge’s contrast of the rest mass vs the field mass is erroneous. The concept of rest mass requires the center-of-momentum frame of reference. Rest mass is indifferent about whether the massive object is a “particle”, a “field”, or both things combined. The problem is that curved spacetime doesn’t admit inertial frames of reference. The only known workaround is to define a reference frame that is inertial asymptotically (on the infinite distance from the object), and this is the way to define masses of strongly gravitating bodies (such as black holes). We can measure lengths, time intervals, and velocities far away of the body, and that’s how we can determine how is it massive.
Also, field energy of gravitation is, in general, very ill-defined concept in General Relativity. One reason is the same as above: it disrupts translational symmetry of the spacetime, and hence hinders the “part of the mass/momentum/energy lies here and another part lies there” discourse. This not only makes a general definition of gravitational energy impossible, but hinders consistent definitions of momentum/energy distribution in a curved spacetime (time is a conjugated to energy, and spatial length – to momentum).
Lynden-Bell and Katz introduce their definition of mass/energy density for some cases. Where the outer space is asymptotically flat, we can unambiguously define the total mass/energy (relatively to this asymptotically flat universe), as well as momentum. But the question “how is this mass/energy distributed inside” does not have a universal solution for several reasons. Should I explicate it? Lynden-Bell and Katz said themselves that “their” mass distribution is different from Penrose’s one.