It looks like the threaded ring that holds the seal inside the pump head has its threads stripped, as it appears to be inside the cover (the "lip" that I marked is what I think is the ring):

If that were the case, you need to remove it from the cover and then you can glue the ring (permanently) on with Shoe Goo, E6000 or the like, or replace it. Have you tried calling/writing Blackburn and see it it can be warranty covered?