We were there Memorial Day weekend through the following week and had no trouble finding pool chairs. You aren't going to find, like, 4 chairs together, but you'll find one to put your stuff on. Beach chairs were no problem at all, got plenty of them whenever we went.
It's hard to see how the Four Seasons will affect cove use, so I wouldn't worry about that. For one thing - the chair areas are separated. They had employees wandering around their chair area just like the CMs patrol the Aulani chairs to make sure their own guests are using them. So either way, no one is going to be going over to the other side. It was pretty obvious that everyone was using the water area most convenient to their resort's chairs. There was also a boat kept between the two resort chair sections some of the time (I don't know boats well... some fancy sailing thing that you can book tours on at that shack on the beach). Other times it was moored on the other side of the Aulani chairs, but maybe there is some reasoning regarding where it's kept.
There weren't enough people staying at the Four Seasons to really make a difference yet, but I don't think it'll matter much when it's at full capacity. There were more locals using the water on the other side of Aulani than there were Four Seasons guests using the beach! Their chair section is smaller than Aulani's, so they must expect to have a smaller crowd. Also, they have cabana things that look over the water on the resort's other side (not the cove, but the ocean proper), so some of the people who are looking to just relax and lay out are going to be away from the cove area.
Locals at Aulani on the weekend make the pool area feel more crowded than the weekdays, but I never thought it felt that bad. and the beach area never felt crowded to me at all.