What resolution is needed to see a quasar through the galactic center?
Whilst awaiting clarification, I'm going to assume that your question is asking about calibrating the reference frame within which the motions of stars at the Galactic centre are measured. 

The process is described by Ghez et al. (2008); does not involve observing quasars at infrared wavelengths; and the study does not suffer as a result.

The Galactic centre adaptive optics images are 5-10 arcseconds across, but contain thousands of stars. Only a fraction of these are close to the Galactic centre. The ensemble can be used, iteratively, to define a relative reference frame by minimising the displacements of thousands of stars, excluding those very close to Sgr A* and those with large motions.

This is really all that is required to do the orbital analyses of the black hole.

To put the coordinates in the International Celestial Reference System, a wider field is observed that contains a few infrared-bright giant stars, that are also bright maser sources at radio wavelengths. The positions of these are used to bootstrap the coordinates onto the ICRS, and this is important if, for example, we are interested in the relative motion of the Sun around the Galaxy.

The maser sources have ICRS positions by virtue of their radio source positions versus the radio coordinates of distant (but still bright and point-like) background quasars.The issue of background sources, extinction and confusion does not arise at radio wavelengths.