Pause Like a Powerlifter – This will make your lifts more honest and make lighter weight feel heavy. If you can bench 300 the conventional way, you'll probably only be able to bench 255 when you lower the weight for 3 seconds and do a full one-second pause on the chest before pressing. Lower absolute loading is a wise choice when you're not lifting with a partner. Save the lifetime PR attempts for lifting days with your buddies.
Use Cluster Reps and Sets – If you're getting fatigued, OR you're trying to lift, say, your 5RM for the day, break the set down into a cluster set: do 3 reps, rest for 10 seconds, and then finish off the last two. It's guaranteed the set will go better.
The rationale behind clusters is to give your body a mid-set, 10-15 second rest to rejuvenate some ATP to give you enough juice to squeeze out an extra rep or two. Use this to your advantage even if the name of the game isn't pushing PRs.
Go To Technical Failure – Training the bench to failure most commonly means performing reps until you can no longer press the bar off your chest. That's where a spotter would come in handy. But without a spotter, going for technical failure is the best way to stay on top of your form and technique while staying safe.
Just stop the set when you know your form is starting to break down. If you feel your hips raising or your shoulder blades "reaching" away from the bench, it's time to call it a set. Don't push it. End the set immediately.
It's not worth risking an injury while you're alone by trying to push past a threshold that didn't need to be pushed. That's training smart.