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 or Google Books.) This book contains a large database of various consequence of choice and tables showing relations between them.
There is also a website 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 which shows implications between various forms we find:
222 142 (1) Pincus [1972c]
142 222 (3) Pincus [1972c]
(1) = "The implication is provable."
(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.
(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.
A proof given there seems to be similar to the proof sketched (very briefly) in this comment.
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 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 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 in this book.