It may be a bit late to suggest that the choice of mass flowmeter may in question, not all mass meters are as sensitive to entrained air as others. Not much point to that argument, however, since i assume the entrained air was not discovered until after the instrument was installed.
Indeed, the mass meter may be a contributor to the problem. Mass flowmeters usually have quite a high presure drop and this may be leading to gas breakout. Increasing the back pressure, provided that it is safe to do so, may give just enough improvement to get you stable readings by either preventing bubble formation or compressing the bubbles enough not to be a problem. It is not simply that the bubbles are in the fluid as they often do not exist is sufficient quantity to give a seriou error to the measured mass and density. However, bubbles can do two things, they can collect on the tube walls which can lead to serious error and instability; and they can introduce what are thought to be "velocity of sound effects". In vibrating density meters the answer is to operate at the first harmonic, instead of the second, where the velocity of sound effects are insignificant. You do lose some density accuracy but will still be well within 1%. I am not sure that mass meters can easily do this.
If not, I can only suggest waiting for other suggestions before you take it out and start again.
