Not that you should do what I do, but I do the opposite. I mean, I don't train heavy squat and deadlift on the same day. Or press and bench press. Or heavy squat and press. On days I squat heavy, I bench press and clean. On days I press I start with light squats (2 sets at about 80% of programmed weight). This means my low back/hips/legs are still fresh for pressing and that's a bigger deal than you might think. Once the presses are done I've still done relatively little in terms of total tonnage so I have plenty in the tank to really attack the deadlift. On days I squat heavy I bench. The fatigue from heavy squatting doesn't seem to affect my bench as much as my press. I finish with power cleans instead of deadlifts for reasons that may not apply to you. My deadlift is quite heavy relative to my squat not to mention I'm old so it takes me longer to recover from a 5RM max effort.
I think the program alternates press and bench press for a reason. If you're training them both in the same workout you're pretty much deciding one is going to be your main lift and the other is just an accessory. That's fine. Like, if you want to really specialize in the press, bench press and variations of it are great accessory lifts. But if you want to push both lifts to your max, I think you need to train them separately.
So for me (an old guy with a relatively heavy deadlift who wants to continue to hit PRs on the press and bench press) my two day split looks like:
Day 1: Squat 5(RM)x3, Bench 5x3, Clean 3x5
Day 2: Squat 5(80%RM)x2, Press 5x3, Deadlift 5x1.
Good luck. Don't quit.