Don't use SSIS for that as it sucks with LOB types.

Powershell or C#  (can be inside a SSIS package script task if you prefer) with a sql reader and a sqlbulkcopy will be the best options.

But it can still be slow to copy - but not as slow as SSIS as it saves each blob on local disk as part of the process.

small example - https://www.codeproject.com/Articles/18418/Transferring-Data-Using-SqlBulkCopy
do remove the notifyafter and the sqlrowscopy bit as it only slows it down