Is it safe to cut and put a compression end on a live cable TV wire?
Cable installers work with "live" wires all the time. There's no voltage on their lines that I'm aware of, only signal and maybe some interference. You do need to disconnect lines for satellite dishes or mast mounted amplifiers since they may have a power injector that adds voltage into the lines.

When crimping the new cable, make sure the center wire is long enough (you can trim it after you crimp a new end) and get some of metal mesh that forms the second wire in tight contact with the crimped fastener. I usually pull some of that mesh back over the insulation so there's a tight bond when it's crimped.