move blob data to new filegroup
You can try moving the data by rebuilding indexes. There are some limitations to this, for example, your clustered index rebuild will have to be done OFFLINE (a limitation of ONLINE rebuilds) and any non-clustered indexes can be rebuilt online as long as the LOB data is not part of the index key or as an include column.

Also, during the index build operation, the transaction log can be backed up and truncated, provided you do not run the index operation in an explicit transaction.

Microsoft has a guide on calculating the space required for here. They also have some general recommendations such as switching to BULK-LOGGED recovery to reduce the amount of transaction log data. See here for guidelines from Microsoft.

One of the advantages to this is you can move an index at a time, so you can do this over multiple maintenance windows if need be.