How do I tell Bedtime that I am awake?
There's no way to get what you're looking for other than what you're already doing.

Bedtime is made-up of two parts: Bedtime and Wake Up.

These are very easy to use but also very limited.  It's not really designed, unfortunately, for one-off changes to the schedule.

I would recommend looking for a custom Shortcut that you can run to turn off the alarm when you wake up early.  You can try Routine Hub and the Reddit Shortcuts sub.