Difference in Backup File Size

No you don't need to install the Ola's solution again since the initially installed commandexecute.sql and databasebackup.sql would have been saved on master DB as SP.

You just need to create a new job , as one created early but this time for 5 left over DB;s and schedule it weekly per you're timings.

EXECUTE dbo.DatabaseBackup
@Databases = 'USER_DATABASES',-- Name of those 5 DB's would come here
@Directory = 'C:\Backup',-- You're backup path
@BackupType = 'FULL',
@Verify = 'Y',
@Compress = 'Y',
@CheckSum = 'Y'

You can use above script while creating the T-SQL job and schedule it accordingly.

More details on how to do above refer to schedule job

Also, just to make sure both, initial backup and this newly created backup job not to run on same time you can put a condition or per the analysis when you're daily backup job completes, you can schedule this new weekly/nightly backup job for remaining 5 DB's

After digging into the backup logs between LiteSpeed and SQL Server I noticed that the first LSN number between the daily full backup and the full backup LSN number in differential backups didn't match up.

After doing some reading on how the LSN values are used in the backups of the database I did some digging on what would cause the full backup's LSN number in the differentials to be different then what the first LSN value was in the full backup on the same day. This post on the Backup Chain in SQL Server pointed me to look at the configuration of the backup jobs and I found that the COPY_ONLY option was enabled for the full backups. (Still trying to determine how it was turned on to begin with and when)

What I learned from the post is that the COPY_ONLY option won't reset the LSN value for the subsequent differential backups which is a big problem if you don't retain the last full backup that was done without the COPY_ONLY option.

From the blog post:

So, stated differently, IF a FULL backup is executed in your environment WITHOUT the COPY_ONLY clause, it can/will jeopardize your backup chain as it will reset the differential base LSN of all subsequent DIFFERENTIAL backups. Meaning, in turn, that if you don’t have (or retain) access to that FULL backup made without the COPY_ONLY option, you won’t be able to recover your databases using subsequent DIFFERENTIAL backups.

I've since turned the COPY_ONLY option off in the backups and the size of my differential backups have dropped back down to what they historically have been. I've also verified that the Full Back LSN number in the logs for the differential backups is matching what the first LSN number value is in the logs for the daily backups.