I'd think it would be hard to determine where the crossover is.
My understanding is the first coil is wound in one direction, the second coil is wound in the opposite direction. When you look at the enamel winding, you will see it's about as thick as a hair. That hair from one coil to the other might not be on the outside of a given coil, so it may be hard to figure out where to cut without killing the pickup. And the coils are most often wax impregnated, so the whole thing looks like a relatively solid mass... I wouldn't want to have to spot where the wire transitions from one coil to another (unless I'm missing something, like the coil wires are brought out to obvious spots for connection after each is individually wound).
Once you've cut, you've got an enameled wire: there's a thin layer of insulation over the wire. You'd probably have to scrape it off carefully with a razor blade. Soldering to new conductors will probably be a pain.
I repaired a small transformer whee the enamel wire parted from the point where it was soldered to the larger leads that you see exit a transformer. That was relatively easy to fix, as the point to solder the coil wire was a fixed spot. I'm thinking you'll have new wires with no great mounting place for them, so soldering the two may be a challenge. And without a good supported spot for that connection to live, it could easily break with handling.
I had nothing to lose when I attempted my transformer repair, as it was already "dead". You might have nothing to lose if you resign yourself to buying a new set of pickups. Or, you could guarantee that you'll be buying pickups, whichever way you look at it.
Just make sure you have a deft touch with soldering, and consider pre-tinning both wires to help the process. Plan on buying pickups, so you have nothing to lose. Hopefully, you have some excellent instructions out there to guide you on how to separate the coils.