How would the Bay of Pigs Invasion turned out if President Kennedy had supplied close air support?

The Bay of Pigs Invasion was a disaster for the fledgling Kennedy Administration, coming just 3 months after he took the Presidency. Its failure is often blamed on the fact that Kennedy did not supply close air support to push back the Cuban Army and let the CIA sponsored paramilitary group get a firm foothold in Southern Cuba.

If Kennedy had committed significant US forces in an unprovoked attack on the Soviet ally, Cuba, how do you think the Kremlin would have reacted? Might they have committed Soviet air and ground forces to repel the invasion, as we did for our ally South Korea when the North and their People’s Republic of China allies crossed the 38th Parallel?