Published books of problems generally mean to nudge serious, dedicated and perhaps talented mathematics students towards a research-oriented frame of mind. If you really mean a bound problem collection for general education, I expect you will have to write such a book yourself. But I would probably recommend against publishing such a book - on the grounds: don't teach until you see the whites of their eyes. A mathematics problem that will work with one cohort might variously and unpredictably either defeat or insult the intelligence of another. And, as the the response to your other question might indicate, teachers of mathematics will have very diverse views of what constitute a value partial knowledge of mathematics.

That said, if I had an audience of highly intelligent but not especially mathematically oriented students, I might focus their "last look" at mathematics on Lawvere and Schanuel's Conceptual Mathematics (which has many good problems). The authors show themselves as both wise and smart. While the book could save the soul of a stray mathematician, it does not harbor any hidden agenda that means ignoring the needs of the broader audience. And while it might accidentally remediate some high school induces confusions, even the best trained students will find most of what the authors say both very new and very fundamental.

Serge Lang's Math Talks for Undergraduate also attracts me, but where Lawvere and Schanuel help a student think about the larger world in a more mathematical way, Lang wants non-mathematicians to understand more about what mathematicians do.

Research mathematician/teachers generally hold as a sacred shibboleth the dictum that "mathematics in not a spectator sport." In the case of a class of general education students seeing mathematics in the classroom for the last time, and and the risk of blasphemy, I question this, I question whether having these students primarily trying to solve problems for themselves necessarily constitutes the best use of their time. I believe that the mathematics community has neglected developing of literature of what one might call proof-oriented spectator mathematics. But still I might recommend a book: I taught a course recently out of Ross Honsberger's Episodes in 19th and 20th Century Euclidean Geometry where I focused on close readings of complicated but elementary proofs of concrete and yet often spectacularly counter-intuitive facts, all material most mathematics majors will never see on the grounds that it isn't sufficiently modern. But as a toy model of what mathematicians do it worked very well for my students.