You are correct that rebooting would be a
very bad idea and would result in permanent loss of your data.
The relevant section of the
Cryptsetup FAQ is 6.10.
Before doing anything else, I would get a printable representation of the master key from the device mapper and keep a paper copy.
Code:
# dmsetup table --target crypt --showkey /dev/mapper/some_name
Typical result, broken into separate lines for readability:
0 200704 crypt aes-cbc-essiv:sha256
a1704d9715f73a1bb4db581dcacadaf405e700d591e93e2eaade13ba653d0d09
0 7:0 4096
Make the obvious substitution for "some_name". The rest of the instructions are in that FAQ section. As long as you have that master key, you
can recover from whatever happens to the LUKS header.