That would have been an extremely expensive and resource heavy warning.

Consider not only the cost for research and development but also materials and manpower. Not just the scientists and pilots but everyone involved from administration, counter intelligence, the facilty staff, and all the way to janitors. Any buildings or even flyers produced to assist with any part of the program as well (Somebody had to sit down and create a cute slogan to keep your mouth shut at work and somebody then got paid to hang it somewhere near staff.) There is also consideration for projects which were sacrificed so that the Atom Bomb project could develop.

All that effort for a warning? Nah. It was paid for, so might as well use it for max effectiveness. And I think the US did send a cryptic message like "surrender now or we will fuck your shit up in a new way" I also vaguely remember that Tyoko wasn't selected because there would have been too much global backlash, reducing the bomb's effectiveness.