A very good reference for various forms of AC is the book Howard, Rubin: *Consequences of the Axiom of Choice*, AMS, 1998. (See [AMS website](http://www.ams.org/publications/authors/books/postpub/surv-59) or [Google Books](https://books.google.com/books?id=ffXxBwAAQBAJ).)  This book contains a large database of various consequence of choice and tables showing relations between them.

There is also [a website](http://consequences.emich.edu/conseq.htm) where you can search for relations between various forms mentioned in this book.

In this book we can find:

> **FORM 142.** $\neg$PB: There is a set of reals without the property of Baire. Jech [1973b] p 7 and note 28.
>
> **FORM 222.** There is a non-principal measure on $\mathcal P(\omega)$. Pincus/Solovay [1977] and note 147.

Then in [the table](https://books.google.com/books?id=ffXxBwAAQBAJ&pg=PA319) which shows implications between various forms we find:

> 222 142 (1) Pincus [1972c]<br>
> 142 222 (3) Pincus [1972c]

(1) = "The implication is provable."<br>
(3) = "The implication is not provable in ZF".

The reference given there is:

> [1972c] D. Pincus: The strength of the Hahn-Banach theorem, Proc. of the Victoria Symp. on Nonstandard Analysis, Lecture Notes in Math. 369, Springer, Heidelberg, 1973, 203-248. DOI: [10.1007/BFb0066014](http://dx.doi.org/10.1007/BFb0066014).

(The correct year for this reference should probably be 1974 not 1973, see the comment below. I left it here in the same way it is written in the book I quoted from.)

D. Pincus writes there that this implication *"is due to Solovay. It is stated without proof in [20]. Since it may interest readers of this paper we include our own proof at the end of §I."* Where [20] is R. M. Solovay, *A model of set theory in which every set of reals is Lebesgue measurable*, Ann. of Math. 92 (1970), pp. 1-58; DOI: [10.2307/1970696](http://dx.doi.org/10.2307/1970696).

A proof given there seems to be similar to the proof sketched (very briefly) [in this comment](https://mathoverflow.net/questions/95954/how-to-construct-a-continuous-finite-additive-measure-on-the-natural-numbers#comment246140_95959).

<hr>

**EDIT:** Since the OP also mentioned Schechter's *Handbook of Analysis and Its Foundations*, I will add that in this book we can find the result in Chapter 29.
In [29.37](https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA810) the following three principles are listed and shown to be equivalent:

 - $(\ell_\infty)^* \supsetneqq \ell_1$; there is a bounded functional on $\ell_\infty$ that cannot be represented by a member of $\ell_1$.
 - There exists a measurable space $(\Omega,\mathcal S)$ and a bounded scalar-valued charge on $\mathcal S$ that is not a measure.
 - There exists a probability charge on $(\mathbb N,\mathcal P(\mathbb N))$ that vanishes on finite sets.
 
In [29.38](https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA810) it is shown that they imply: **(NBP)** There exists a subset of $\{0,1\}^{\mathbb N}$ that lacks Baire property.

As far as the references are concerned, the author says that:

> This implication was first stated without proof in Solovay [1970]; the first published proof apparently is that of Pincus [1974]. The slightly shorter proof below is essentially that of Taylor; it was published in Wagon [1985].

The first two references have already been mentioned. The third one is S. Wagon, *The Banach-Tarski Paradox*, Encyclopedia Math. Appl. 24, Cambridge Univ.
Press, Cambridge, 1985. The corresponding result is indeed given in as [Theorem 13.5](https://books.google.com/books?id=_HveugDvaQMC&pg=PA211) in this book.