The reed mats should indeed be fastened with wire at reasonably even intervals across, as there is a risk you might cut the wires and cause a sag/drop.
The question is whether you should use a router, a tool or machine that is cutting or finely serrated (here some might suggest a Multimaster) probably causes less damage. Make two parallel cuts and clear between them.
If you make an attempt, you should be prepared that in the worst case, you might have to put up plasterboard on the ceiling, but what doesn't one do to avoid ugly exposed cables...
Best regards,
Ronnie