As one of the Addgene scientists responsible for Zhang lab web pages, I just wanted to make a quick clarification on those primers. Since there is no gRNA in the empty vector, there are no sequencing primers suggestions for that region. The primers you see are for the gene inserts that already exist within the plasmid: Cas9 and Puro. If you click on the "View Sequences" link on the lentiCRISPRV2 page, you'll see that we also sequenced the gRNA insert region with an hU6-F (5'-GAGGGCCTATTTCCCATGATT-3') primer. As you mentioned, LKO.1 will also work.
I cannot remove or change those primer suggestions because, technically, they are correct. The constraints of our data entry form and the nature of this plasmid render the information a little confusing, but it's not actually wrong. I am going to add a note in the comments section of the plasmid explaining what primer to use to sequence the BsmBI region so scientists can check their own gRNA inserts.
Best,
Eric