In the Bay area I'd think heat stress to be rather rare. Any stress they show now is likely because they were just planted and maybe too dry. If they continue to show heat stress after well established put some shade cloth over them. Probably about 40-50% shade.

Can't speak to the blackberries but my 3 raspberries (Caroline, Purple Royalty, and Latham have no problem at all with full sun and temps. in the 90s with howling wind. My berries are on drip too, perhaps you are over watering them and killing roots, that could give the appearance of under watering as the leaves would wilt back and in the heat from dead roots. Just a thought. Make sure and observe the wetting pattern in the soil when you irrigate to see where the water is going, I put 2 lines down the bed to cover the full width of the bed with each irrigation.

Are the upper leaves or lower leaves the ones browning, is the browning over the entire leaf or just the leaf margins?

Thanks for the follow ups. I thought it may have to do more with the fact that we just put them in.
I have one 2 gallon/hr nozzle for each bush, but its at the end of the line so I'm not sure if it had been getting that much water. Also, prior today, I'd just been running that drip line for 20 min a day, so I think it's clear that I need to increase that. I also plan on adding additional nozzle to make sure I get full coverage of the planted area.
The browness I observed seemed to be originating in the middle of the leaves, not the margins? I didn't notice if it was more upper or lower leaves, I'll check that out when I get home today.

I'd bet as described that the amount of water for the current temperatures that you are experiencing for the blackberries, are not enough. Blackberries like to be well watered. Just as an example I water for 90 Minutes every other day on my 33 blackberry/boysenberries (1 gph). Just an FYI, but raspberries need quite a bit less water.