I'll go against the grain here and say you should not do it. You should not even consider it. Your response to the professor was spot-on.
I'm assuming that you and some of your classmates will meet the 14/20 requirement after finals, so I'll discuss only the 160 credit completion requirement. By setting the credit requirement for payment so high, the university has made it difficult to find TAs who could be paid. Why?
On the charitable assumption, the university wants TAs who are close to graduation because being a TA is a substantial commitment; for those who still need a lot of credits, the time spent as a TA could potentially impede their progress toward the degree. In that case, take the university at its word: since you do not yet have the number of credits, focus on completing your degree, not on TA work.
On the uncharitable assumption, the university doesn't want to pay TAs, and asks that TAs be close to finishing their degrees both so that few students qualify for payment, and so that those who do qualify will soon graduate and leave. Since the professor has offered you the job, to my mind this is more likely than the charitable assumption.
There could be a neutral assumption that those close to finishing their degrees are more qualified to be TAs, so they deserve payment whereas someone farther away from finishing is less qualified and should work for free. I'm cynical enough not to believe this. After all, the professor offered you the job. Why would he do that if he thought you were not qualified? And if you are qualified, why not pay you?
If the requirements for payment are out of his hands and he is simply abiding by rules, he needs to realize that the rules are unreasonable and to take that up with the powers that be. He should make the case that it's not necessary for TAs to be so close to completing their degree requirements. If he doesn't want to do that, he shouldn't be asking for students to work unpaid. Asking you to accept the unreasonable situation is not fair to you.
Finally, most of the "go for it!" answers here are hand-waving away the actual work. It is not trivial to prepare four class meetings per term. Exam prep classes still require you to be as fully prepared for the exam as if you were going to take it yourself; to be able to explain complex material very succinctly when a student has a question, since the exam (presumably) covers several weeks' worth of material; and to deal with students' anxieties. That is hard. Academia tends to undervalue teaching, and to underestimate what it takes to be a good teacher—note how few questions on this site are about teaching rather than research, or the politics of doctoral programs, or submitting/reviewing a journal article, etc. Assuming you're conscientious and want to serve your students well, you will spend a lot more time on your TA work than it seems.
I don't know what country you are in, but in the US, academia is notorious for its poor treatment of front line workers who do the bulk of teaching, such as adjunct faculty. Your situation is not that different: unequal pay structures that both perpetuate and result from an inherently exploitative system. Those who want an academic career might heed the "intangible rewards" discourse of the majority of the answers here. But if you don't have a political need to be on the professor's good side, why should you buy into it?
Full disclosure: I have been a TA, both paid and unpaid, at various universities both in India and the US. Any time I took an unpaid position, it was for one of two overlapping reasons: I wanted to (re)learn the material myself, and/or I wanted the affiliation with a well-known mentor at a prestigious university. Neither of those considerations seems to apply in your case, so there is no reason for you to do gratis what someone else would be paid for. Do not accede in your own exploitation and prop up an unjust system.