.... They can't take the island...
Yamamoto earmarked 5,000 troops to take on 3,000 top-quality defenders. At sea he committed to the venture something like 6 carriers, 11 battleships, dozens of cruisers and destroyers, supported by about 15 tankers, (call it about 120,000 tons of oil available for extended operations).
So the US carriers are sunk on the 4th of June. Let's say Kusaka did the dual phased search and spotted the US carriers at 5am, so quickly that even the Midway strike was successfully diverted. Whatever. Now, the land invasion is going in on the 5th backed by corps level artillery support. Can this attack take both islands? No, certainly not. Can it take one of the two? Unlikely, but maybe. Can it attain a foothold on one or both islands, but then get bogged down? That's the most likely outcome.
If so, on June 7th Yamamoto is looking at a bridgehead on one of the two islands with the defenders holding fast and the attackers spent and reverted to the defensive. If so, his options are either (a) evacuate or (b) reinforce. If reinforcing, then what he would do would be to land more 'troops' from his warships (ie, shore parties) to bolster the bridgehead, then arrange for a brigade sized IJA force to come to Midway, perhaps for around the 20th of June. Since the US carriers are sunk he would not need to maintain a large fleet near Midway and would strip it down to a carrier force, backed by ample tanker support so that it could blockade Midway until the (scratch) invasion force had arrived 2 weeks later.
That's the 'reinforce' scenario, the premise of which is that if the invasion has a bridgehead the side that can reinforce will take the island. The 'evacuate' scenario is that Yamamoto on the 7th decides the island isn't worth the trouble, or the IJA says in response to the navy's plea that they're not sending a brigade to Midway. Either way, the key is sea power allowing the side with it to choose what to do, and your premise is that the Japanese have it.