You should be able to use either of the third-party applications "Carbon Copy Cloner" or "SuperDuper" (free for this purpose) to copy your case-sensitive volume to an empty case-insensitive one. Make at least two such copies on different drives. One is not enough to be safe.
If there are any name conflicts—that is, files in the same folder with names that differ only in case, such as "File" and "file"—then you will either get an error or one of the files won't be copied. You must ensure either that no such conflicts exist, or that the consequences are not important. How you do that is up to you. Unless you went out of your way to create conflicts, they probably don't exist.
Then erase the source volume in Disk Utility as case-insensitive. This action will remove all data from the volume. If the volume is the startup volume, you must start up in Recovery mode or from another drive in order to erase it.
Restore from one of your backups using the same application you used to create it, or use the "Restore" feature of Disk Utility, which will be faster. Search its built-in help for the term "duplicate" if you need instructions.