I googled ... and didn't really find anything to directly solve your problem.
However, I did find several recipes where you DO NOT cook the beets. Rather, you slice on a mandolin or grate the raw beet. Put in jar and add the hot brine. Apparently makes a nice crunchy pickle.
With that in mind? Could you slice the beets? Or chop? and eat them in their crunchy state? Or reboil their own pickle juice and add to the jars?