All is not lost! Find a nib that is attached to the feed assembly, like that of Lamy's, wrap some plumber's tape around it until it forms an air tight fit when inserted into the body. The only air exchange should happen only through the feed assembly and not around the edges.
Do note that all feeds assemblies are not equal! The Varsity body is huge and when the ink is almost finished, heat from your hand will cause the air inside to expand and overwhelm the collector fins; causing burping. Reduce the air capacity by stuffing beads or small foam blocks into the body.
The video here explains how to replace the nib/feed assembly and refill it from then onwards.
Remember to save the nib! It can be used by other Varsity or Pilot Petit1.