Your storage can be cheap/large/fast. Pick two. If you’re looking at backups since he’s operating on a local SSD, does it really need to be fast? Even if you were pushing an LSI 9207-8i (though you’d need the 8e), it will take close to 14 minutes to transfer 4TB. (Sata/SAS controllers tested (real world max throughput during parity check) - Storage Devices and Controllers - Unraid) So, yeah, DAS is your best contender, and remember to price out the card you need to put in the video editing machine.
The hardware to do the DAS without the hard drives could probably be found cheap enough. Maybe 30-40% of your budget. The hard drives are what’s going to tank your budget, especially if you want to back up more than one project.
Here’s a different thread of thought: Does the entire 4TB project change at one time? If not, only back up what’s changed and that will save you a massive amount of time.