Migration of file server data to OneDrive (Office365)

I have the following question and hoping someone can answer:

I have to migrate data (multiple terabytes) from a Windows-based file server to OneDrive (Office365). The data contains archived data of the last 10 years. Commvault Archiving is being used. Currently, the single policy archives all files which have not been accessed for the last 5 years.

I can migrate data, but the archive files (stubs) are unarchived as soon as I migrate the data to OneDrive. This caused file server space to bloat as the stubs got unarchived.
The client does not want to delete any data even after migration. 

I have the following question:

How to mitigate the risk of the file server going out of space during migration?

Also, please advise any other migration strategy to OneDrive other than direct copying, which does not risk file server space bloating.

Thanks.