That's a very good question.
I think the shortest route to joining the East Asia Summit is to engage with East Asian states, particularly the ASEAN states. ASEAN drives the pace of regionalism in East Asia. The way into the East Asia Summit is to engage ASEAN, not so much China, Japan, and South Korea. ASEAN has the normative veto on who gets in and who doesn't.
With that understanding, very few ASEAN states, with the exception of Singapore, have made an argument about becoming Arctic states. I don't think that being conciliatory or welcoming on the Arctic Council side advances us very far in the East Asia Summit.
The ASEAN chair was clear that what he wants from us is more regional engagement. Whether or not we could trade Arctic Council support for support in the EAS is a separate question. Possibly; I don't know. That might be worth exploring, although we'd be getting more than they would because a permanent observer on the Arctic Council doesn't get to do or say much, whereas as a member of EAS, you do have a bit more say.
I'll make one quick point on the South China Sea. It's not worth comparing China in the Arctic and China in the South China Sea. China has a territorial claim in the South China Sea, making the South China Sea a priority for the Chinese government. The Politburo Standing Committee does not discuss the Arctic, so China does not perceive the Arctic and the South China Sea through the same lens.