I believe the old "book rate" is gone, and now the closest equivalent is "media mail".

You may get away with it, but I don't think magazines qualify. Here's the section from the USPS site:

4.0 Content Standards for Media Mail
4.1 Qualified Items
Only these items may be mailed at the Media Mail prices:

a. Books, including books issued to supplement other books, of at least eight printed pages, consisting wholly of reading matter or scholarly bibliography, or reading matter with incidental blank spaces for notations and containing no advertising matter other than incidental announcements of books. Advertising includes paid advertising and the publishers' own advertising in display, classified, or editorial style.

b. 16-millimeter or narrower width films, which must be positive prints in final form for viewing, and catalogs of such films of 24 pages or more (at least 22 of which are printed). Films and film catalogs sent to or from commercial theaters do not qualify for the Media Mail price.

c. Printed music, whether in bound or sheet form.

d. Printed objective test materials and their accessories used by or on behalf of educational institutions to test ability, aptitude, achievement, interests, and other mental and personal qualities with or without answers, test scores, or identifying information recorded thereon in writing or by mark.

e. Sound recordings, including incidental announcements of recordings and guides or scripts prepared solely for use with such recordings. Video recordings and player piano rolls are classified as sound recordings.

f. Playscripts and manuscripts for books, periodicals, and music.

g. Printed educational reference charts designed to instruct or train individuals for improving or developing their capabilities. Each chart must be a single printed sheet of information designed for educational reference. The information on the chart, which may be printed on one or both sides of the sheet, must be conveyed primarily by graphs, diagrams, tables, or other nonnarrative matter. An educational reference chart is normally but not necessarily devoted to one subject. A chart on which the information is conveyed primarily by textual matter in a narrative form does not qualify as a printed educational reference chart for mailing at the Media Mail prices even if it includes graphs, diagrams, or tables. Examples of qualifying charts include maps produced primarily for educational reference, tables of mathematical or scientific equations, noun declensions or verb conjugations used in the study of languages, periodic table of elements, botanical or zoological tables, and other tables used in the study of science.

h. Loose-leaf pages and their binders consisting of medical information for distribution to doctors, hospitals, medical schools, and medical students.

i. Computer-readable media containing prerecorded information and guides or scripts prepared solely for use with such media.

I think the "advertising" prohibition is what would rule out the Club magazine, even if you call it a "book".