I definitely agree with squatting more than 1x per week. I'm just trying the BBB approach now (adding BBB squats to dead lift day), haven't run with it long enough to know much but I'm sure it's better than what I was doing.
On presses I think you're on the right track to drop the volume on the BBB part.
That, or you might consider just alternating which BBB part you do each cycle (or week):
Odd Cycle (or week):
5/3/1 Bench + 5/3/1 Press
BBB Bench
Even Cycle (or week):
5/3/1 Press + 5/3/1 Bench
BBB Press
This template also alternates which movement gets the benefit of being done first if that's worth anything. I think I would like alternating every 4 week cycle better than weekly.