Recording 4k with Video Assist 7" 4k with non-4k camera

Can you record 4k onto the Video Assist 7" 4k with the original BMPCC which isn't 4k itself?

No, not yiu can not. The Pocket camera (original model) is an HD 1080 only camera, so you can not get a UHD signal to the VA. You need a UHD/4K Camera to record any resolution higher than 1080p.
That said, the VA 5-inch model,pairs nicely with the BMPCC as does the new SmallHD Focus Monitor, which will also power the Pocket Camera.

If you want to record UHD/4K, then order thr new Pocket Camera 4K.

If you did a topic search here, you would see this has been discussed before.
Cheers