The short answer, unfortunately, is "no".
Now for the longer explanation.
Quick Base will look at the expected volume across all subscriptions/reminders when you set yours up to find the least utilized time for performance reasons, and insert it into the queue for a specific time. As you noted, that typically means sometime in the middle of the night. From that point forward, it will trigger at that time of day each time it is sent.
The only trick I've found is to setup 10 or so subscriptions with a title that helps me identify which was sent when. After each triggers tomorrow morning, I'll delete the 9 furthest away from the time I want and keep the one I need.
------------------------------
Eric Mohlman
------------------------------