Yes. The DAC7611 can accept a clock up to 20MHz (50ns). I doubt that it's being used that fast, but who (other than the OP) knows.

Potentially that doesn't give you much time between the first and second bytes, but since the second "byte" is only four bits long it won't set BF anyway so I'm not sure how you'd get at it/know when it's done being transferred.

Also, with CS wired low, you'll have to come up with some way of using the LOAD pulse to resync transfers if you ever get out of sync with the transmitter. Otherwise, you'll never know high bytes from low bytes.