I should have checked beforehand, I always think Huntingtons is recessive, but it's dominant. In which case your own comparison of the Pearl to Huntingtons is wrong. If it is in fact caused by the Woma gene then it's a recessive disorder, not a dominant one.

There is still a chance that the fatal part of the Pearl is caused through genes that have presented themselves through line breeding and we just haven't been able to get the two separated (they could be close to eachother on the same chromosome and thus may never have crossed over), in which case breeding the line out would be effective at producing a Pearl that would thrive. However it is possible that the Woma gene itself is what is causing this issue, in which case the two will never be bred out.