Thanks for your reply, and for confirming that the shortcode terms are supposed to carry over to the user search. I just started testing, and will report my findings.
Setting up the plug-in at a different site brought up something that confused me months ago when I first started using this plug-in, which I'd set aside. I don't mean to waste your time, but I think the problem may turn out to be connected, and might even lead to something useful for you and other users of your plug-in
The SUL search is initially set up to "Search authors by name." So, if we add SUL on a site with an author named "Miller," and we search for "Miller," we'll bring up the author or authors named "Miller," with links to all posts by however many Millers.
To my understanding, this is how it's supposed to work, and getting the search to search for anything else ought to require creating a custom search function, as in the "billing_city" example you use under your FAQs.
Interestingly (and usefully!), for reasons unclear to me, and without my having done anything (intentionally), on the site where I first installed the plug-in, if I do a user search for "Miller," it will bring up all of the authors who mention "Miller" in a custom field added via the membership plug-in S2Member and stored in the usermeta table.
I'll add that I was delighted with this to me unexpected functionality, as I thought it was something I was going to have to figure out how to add later on. I also expect that I will need to understand exactly why it's happening in order to gain control over types of searches and outputs, since eventually I'd like to able to give users a wider range of options for narrowing searches and/or results.
Here's the page - it's a site in "Beta," but I hope for not much longer - where the initial list DOES employ the shortcode parameters, but a search DOES NOT. On the other hand, it DOES search the usermeta table and a specific field within it without my ever having specifically "asked" for that functionality - you can test using "Miller" or "Ashtanga," for example:
The YIR - Listings
You'll see that in my customization I added a note on searching the entirety of the user "profiles" at this site. I'll just need to figure out how and why to get the further control I want, and to avoid bad surprises (i.e., the day that functionality just disappears).
A suspicion is that there is some unexpected fraternization between your plug-in and S2Member that produces the custom field, but it's just a suspicion at this point, and for various reasons somewhat laborious to test.