How can I set up a secondary permalink structure?

I think I found a solution to this with a function that’s not commonly used: add_permastruct().

This does the trick of what I described above:

add_rewrite_tag('%page%','([^/]+)', 'pagename=");
add_permastruct("abc','/abc/%page%/',false);

add_rewrite_rule('abc/?$','index.php?page_id=6','top');

The add_rewrite_tag() defines the pagename value that I want WordPress to lookup. Then add_permastruct() defines my custom structure with that value.

The last rule is just to handle the main page manually.

I believe this would only work with one-level of hierarchy, but I may come back and tweak the regex later to improve it. For now, this works for my purposes.