Patrolling shipping lanes is a peacetime operation, so I don't see how the First Island Chain matters. They're not going to halt Chinese naval ships going on patrol missions. It just means the patrols won't be secret.
Patrolling shipping lanes is a power projection, one that US allies and non-allies alike enjoyed or tolerated due to the demonstration of the US’s impartiality and commitment to free trade. China projecting such power will not be seen as impartial, especially given the never-resolved territorial disputes in the region.
