I think that the problem is that if you pick the first block mined after April 1st midnight, and there are two competing blocks, it will be a problem. In the blockchain one of the blocks will win sooner or later and it will be extended with a long enough chain to kill the other block. But the people that would have won the lottery with the other block will be angry.

And remember that the timestamp of the block is oply an approximation. You can use any time you want. I'm not sure if the net reject the block when the time is too off, but you can use a fake delayed time of a few minutes and wait to broadcast it (perhaps mining secretly the next block), or perhaps you can use a fake previous time of a few minutes and blame the delay of the broadcast to a bad connection. (It's mode difficult to do this secretly in a pool.)

The newspaper here make a big fuzz about the first baby of the year, and I always suspected the exact born time in that moment is not reliable.