The problem at the moment is that it is exposed to the oxygen in the head space, as a result acetobacter is rapidly multiplying. If I were you I’d get it into a smaller container and refrigerate until you use it. You can dry the oak on a rack and then bag it up for later use, this has given me good, but not great results (sometimes I’ll just toss oak directly into another sour when I bottle a beer).
I repitch primary sour yeast cakes pretty regularly. For secondary yeast cakes I sometimes will pitch about a cup of slurry along with a fresh pitch of healthy Saccharomyces and additional bugs.
Hope that helps, good luck.