I’ve worked with this exact setup before, and the key thing to understand is that RoboCopy cannot pull tiered files directly from Azure. If you point RoboCopy at a tiered path, it will try to recall the file to the local server first. Since you don’t have enough space, that won’t work.
Here’s the practical way to do this with the least pain:
Use Azure Data Box for the bulk export. That gets you the 300 TB without needing local recall. This is the right first step.
After the Data Box seed, switch to AzCopy for the delta. AzCopy reads directly from the Azure Files share, not from the on-prem cache. That avoids tiering recalls and avoids running out of space.
Keep RoboCopy only for the final small cutover window. Use it from an on-prem Windows box to capture the last few locked files and permissions right before users are moved. By this point, the volume is small enough that recall is manageable.
So, in short, "Data Box" for the bulk, AzCopy for the live delta, RoboCopy only for the final tiny change set. RoboCopy cannot bypass tiering, AzCopy can.