Does this sound logical?

No, but in my experience people do speak like this.
e.g. "I'm broke, but I can still feed them."
Also I think that sometimes people use it to mean that they have no money in the bank, even though they might have money in physical form.

He might have bought it using his credit card. Buy now, pay later.

Note that お金ない is in the present tense. The speaker had little money when s/he bought a/some book(s), and still has little money now.

ない sometimes means you have only a small amount of something.
筋肉がない, 胸がない, 音感がない, 語彙力がない, ない頭を絞って etc.

Maybe you can use "I'm out of money" or "I'm broke".

Do you think that maybe we should delete this sentence now, or change it to one of the suggestions?

Could it be amended to:

didn't ... I bought another book
didn't ... I bought some more books
didn't ... I still bought a book
didn't ... I still bought some books

?

It would then sound natural.

I'm not sure if that matches the Japanese. Does it mean that the speaker is still short of money when s/he's speaking?

The current sentence with "don't" means that the person doesn't have any money at the time of saying this sentence.

I don't think the second part of the sentence using the past tense "I bought" sounds natural because I don't think we mix present and past tenses in this way. That's why I agree that the "don't" be changed to "didn't". With "don't" I would say something like "Even though I don't have any money, I'm going to buy..." (as discussed earlier, it's a bit illogical but grammatically correct).

"didn't" clearly means that the speaker didn't have money at a past time.

The situation is, for example:
I had only 1000 yen. I bought a book (or some books) and spent 600 yen. Now I have only 400 yen.