How about Sigma Xi research society? That is a great one if you can have a full membership. There is a discount dues for students. Ask your freinds of Sigma Xi membership to recommend you to join.
I have no idea about APS' bylaws. Look at the bylaws of any potential societies of your interest. If the bylawys say the membership are assigned based on certain selection criteria. That will be a great one to consider.
If you get any invitation letter from any societies, the letter will be a great evidence to show your impact or expertise even if you decide not to join it. Good luck.