First step is to learn how to predict this situation and then use proper cutting technique to prevent split from happening. I will not begin and hope no one else attempts to explain those techniques here in print. You should have that shown in real life. I will share one alternative method of tying in however. NEVER rely solely on that flip line/lanyard. In addition, you should be tied in with your climbing line to a nearby tree, or co-dom leader. If you get injured or run into mean squirrel or bees or hornets, handy to have that climbing line already tied in and ready to descend on.
When all I have is the very spar I am piecing down, I have the ring type friction saver around that spar near my knees or feet with climbing line through rings making choke hold on tree. If any possibility of a split exists, just un-clip flip line from one side of saddle and clip both ends of flip line to the same ring on saddle. You can't get squished, and if that "circle" breaks, you are still tied in with climbing line, ready to descend to mother earth and clean out pants.
PLEASE have this properly demonstrated before trying out on your own?