With the assumption that running a full integrity check on the primary, Live instance is not really an option here, my approach would be something along these lines:
- backup of your primary instance using your preferred third party tool
- automate the restore of the backup into a separate environment
- run the integrity check in this second server
I see a couple of advantages in such a scenario:
you can verify that your backups are indeed valid and can be restored, quite a good felling in case of actual needs at some point in time.
As an integrity check is quite I/O intensive, you would move all those extra IOs away from your Live storage and only affect a separate set of disks.
Our documentation includes a section on how to script integrity checks:
"Checking Database Integrity Using the ^Integrity Utility"