A couple thoughts that might help...
-- Use a trim removal tool to start to pry up the hatch in areas, giving better access to those areas that need cutting, still. A trim removal tool is basically a plastic pry bar. Some are stiffer/stonger than others.
-- A metal pry could be used - a thin piece of metal like a flexible putty knife can help protect.
-- If you haven't, obviously a SHARP razor knife will work. You can get plastic razor blades, if you want.
-- Depending on how steady/confident you are, you could use a rotary or oscillating cutoff tool. An osc will likely be the easiest to control. These will heat up the caulk so do it in spurts or take turns at different spots.
-- A heat gun may help soften the material, making it easier to cut. Too much, though, and it will likely just melt back together.
-- When you re-seal, you really don't need a super-strong adhesive caulk. Any medium-strength is MORE than enough.
-- Bury a thin, strong twine (or heavy fishing line) in the perimeter before adding the new caulk. Don't overdue it with the caulk. We you remove it again, dig out one corner and then simply pull out the twine.