That fish and many of my others are caught in very bad areas for line cuts. Blasted shale rock covered in Zebra mussels cut any braid with the slightest amount of pressure.
I use a 40 foot shock leaders made of 80 lbs mono. Not ideal to work with, nor for distance casting, but you get used to it. Tie it to your braid mainline using the "FG Knot", best one I've used to date, just make sure it's done right and pull as tight as you can to test it before. Youtube video is at:
https://www.youtube.com/watch?v=pjzUb5QRKuk For hair rigs, I use 50 lbs power pro with size #4 curve shank type hooks. Either Kamasan b725 or 775, or Fox Arma point SSC. Hair length distance from leader is roughly 12 to 15 inches. When I need to use popups (fishing over decayed weedbeds), I adjust my length as needed, staying as close to bottom as possible.
I make my own boilies since years. No preservatives, I freeze them until use otherwise they go moldy within 5-7 days. Mix of semolina, corn flour, soy flour, some seeds and dry flavours like hazelnut creamer, chocolate, cinnamon, peanut butter, garlic, paprika, fishmeal, just to give you some ideas for flavours.
If your hook keeps popping out of the carp's mouth on a run, you may be setting the hook to hard, assuming your rig properly tied and rigged. Boilie shouldn't be to close to the hook, nor too far. 1 to 1.5 centimeters is about right, if your boilies are different sizes, adjust your hair length accordingly. I do sell some hair rigs on my web site if you want to try them, see:
http://www.freshwaterphil.com/hair-rigs.cfm
