Database Mirroring setup does not work when the database is above a certain size, logs out of synch
As has already been noted there is likely to be a third party tool taking backups which you weren't aware of.  Stopping that (or at least having control over it) is going to assist no end (obviously be aware that stopping log backups may invalidate the RPO/RTO targets and taking adhoc full and log backups will invalidate the log chain and may make dr recovery more complex or result in larger data loss during this period).

You've mentioned that you are keen to reduce the downtime required; I believe you have a few options available to you to assist.

I note that you state SQL Server 2012.  Backup compression is available in standard edition upwards, enabling this will reduce IO (and therefore time) for both your backup and restore at a relatively small cost of CPU.  I have seen backups reduce by nearly half using compression (depending on what is stored in the DB).  Simply add compression to your with options.  This can be used in conjunction with the below to reduce the outage window further.

Option 1 - Use a differential backup

Rather than taking a full backup at your outage time, take a full backup (or last nights if done daily) and restore that during the day of migration to your other servers.  You could then take a differential at your maintenance window.  As a differential backup only contains the changes since the last full backup it will be much smaller and faster than a full backup.  It also doesn't have any dependency on log backups so as long as your full backup is good you can apply a differential to get it in sync.  Be aware though, the differential must be taken before the next full backup so either know that the planned window is before the nightly backups or suspend the full backup for this database.  

Therefore you could:


Take the full backup on A and restore it to both B (NORECOVERY) and C (NORECOVERY) with the application still up (and log backups continuing).
Wait until your outage time, stop log backups and take a differential backup
Update your connection string (or CNAME)
Restore differential to A (with RECOVERY) and B (with NORECOVERY)
Initiate the mirror between B and C
Start log backups on B


Option 2 - Mirroring

As you are not in a mirrored configuration currently you could look to create the mirror between A and B first; even if going up SQL Server versions mirroring will still work one way (from older to newer versions).  Restore and mirror would be done with the application still online, without requiring downtime.  

This might look something like:


Backup Server A and restore to B (NORECOVERY)
Stop log backups on A, take differential and restore to B
Initiate mirror (A --> B)
Restart log backups
When your outage window comes up, stop the application services, change the DNS CNAME (or connection string)
Failover the mirror (from server A alter database [mydb] with partner failover;
start the application services again
Start log backup schedule on B
Backup full db on B
Restore to C (NORECOVERY)
If log backup has been taken in time between backup and restore from B to C then take differential on B and restore to C
Initiate mirror


Option 3 - Log shipping

By utilising log shipping you can actually seed both B and C simultaneously.

The process would be similar to below:


Take full backup and restore to B and C (NORECOVERY)
Stop log backups (permanently on third party tool)
Take and restore differential to B and C (NORECOVERY)
Start log shipping configuration with 2 secondaries (B and C, NORECOVERY).  This will be creating the log backup files so may need to mention this to the backup guys so they capture the location, note that the SQL Agent services will need to have read/write permissions over the share(s) for this to work.
When downtime outage comes, take final log backup and restore to secondaries (jobs found in agent for backup/restore jobs on respective instances)
Change CNAME (or connection string)
Bring server B database out of restoring state restore database [mydb] with recovery
Initiate mirror
Start log backups on Server B