So, if we where to accept your premise that the White House knew not only
that the Japanese where coming but also the day they would be "arriving",
why not warn Pearl Harbour?

Good Question,but I think Kimmel gave a very clear answer to this question
during interview in 1958.

After all, if you're planning on joining a
war it's generally considered an advantage not to lose large amounts of
men and equipment on day 1!

Amounts of men and equipment lost in Pearl Harbor is very insignificant in
comparison with manpower and production resources of US,as the later
developments in war proved.

No need to sortie the fleet if you still want a "sneak attack", just get
them placed on alert - ie: all anti-aircraft guns manned and ready and a
decent CAP (with the rest of the fighters on +5/+15) overhead.

Pearl Harbors shortcomings were well known,in fact during pre-attack meetings
Kimmel always maintained that only viable defense would be the keeping the
fleet in open sea and his views shared by everybody.