If that's the policy, places around here don't seem to follow it. I almost always print my labels late at night (and don't touch the ship date field), and drop off the packages the next morning.
about half of mine go into a drop box (they can't really deny those
), the other half to the counter at the local UPS Store, and they've never said anything about it, other than occasionally double checking the weight.
Maybe worth trying other local places that accept USPS to see if they hassle you?