The other response is incorrect because Operation MI was not intended to draw out the Americans and destroy them-at least, not initially.

The entire operation was a two-stage operation: first, the Japanese would capture Midway Island, through a combination of airfield strikes to destroy the aircraft present on the island, followed by a bombardment fleet of cruisers and then the main landing force. Once captured, the Japanese hoped that the fall of Midway would force the US to commit ships to retake it. Through use of a submarine screen and floatplanes based off Midway, it was hoped that they would be able to detect the incoming Americans, and then find a weak spot and smash them-in effect, the exact opposite of what happened in the contemporary Midway battle.

The reason the fleet was split so awkwardly is due to the battle line configuration of the Japanese. The First Air Fleet (Kido Butai, or KB) was-despite Kaga's sluggishness-a "fast" fleet, hence its name, the "Mobile Squadron." Simply put, they were faster units than the rest of the main fleet. While placing the carriers with, say, the Yamatos, may have given them enhanced anti-air screening (through the theoretical deployment of the Yamato's Type-3 antiair "beehive" shells, as well as the other escort and capital ships), the problem is that the Yamatos were simply too slow to accompany the carriers. The carriers would need to crawl along at the same speed as the heavier battleships, which would delay the operation and give more time for the Americans to react. The main fleet was not initially thought to be needed immediately, as the actual battle for Midway was intended to take place after the capture of Midway.

The "Invasion Force" was a mixture of a screening and a bait force. The Japanese, despite the fact that they would lose a war of attrition, loved deploying "bait" for an enemy to attack. It consisted of a variety of the Kongou-class battlecruiser/battleships as well as some heavy cruisers. The purpose of this force, aside from executing the actual landing, was to be a target for any potential carrier raid from the Americans, the idea being that the American carriers, seeing the presence of two large capital ships, would presume it to be a target worth attacking and commit their naval aviation against it. This would then allow either KB or one of the other formations to respond.

When you say CruDiv I assume you mean CruDiv 7 (Mogami, Mikuma, Suzuya, Kumano). Their role was to bombard the island in preparation for the invasion. Disregarding the fact that 8"/20.3cm guns would be of questionable effectiveness against hardened Midway defenses, their job was to essentially "dash" in and open fire. As such, they weren't attached to any of the slower formations, because that would slow them down.

The primary purpose of the battle of Midway was to entice the main body of the US fleet into a decisive battle and then destroy them. Yamato spread out his forces to disguise the true strength of the fleet amassed around Midway, which would encourage the US to commit itself to defending the island.

They would then be fixed and engaged by the Kido Butai while the rest of the fleet would surround and destroy the entrapped Americans. At the time, the US fleet was in a weakened state and Yamato wanted to prevent it from licking its wounds and recovering, by finishing it off completely. The plan also intended to have a screen of submarines to harass the US fleet, but there were delays in establishing this and submarines also proved to be far less effective at engaging main fleets than naval theorists in the 1930's had hoped.

Unfortunately, the Kido Butai was functionally wiped out within a day and this removed the ability of the Japanese fleet to fix the Americans in place, as well as severely damaging their striking potential, forcing the Japanese withdrawal.

Source: Smith, Peter Midway Dauntless Victory; Fresh perspectives on America's Seminal Naval Victory of 1942. Pen and Sword Maritime, 2007.