I agree with Rhodium.
I'm not sure the mixed anhydride will be stable. It may well disproportionate to the respective non-mixed anhydrides; this would explain why there is only a single reference for 3-phenylpropionic acetic anhydride in Beilstein, and that is a kinetic study: Indian J. Biochem., 13 (1976), 350.
There will probably be little chance of regioselective nucleophilic attack as the carbonyl groups are too similar to one another.
I know you don't like acid chlorides but the 3-phenylpropionyl chloride won't be particularly volatile, unlike the stinky lower order acid chlorides. If you could use the crude acid chloride as Rhodium suggests this would minimise your exposure (admittedly the use of thionyl chloride may negate any benefit from this).
An alternative is the use of trifluoroacetic anhydride to make stable mixed anhydrides which are entirely selective for attack on the preferred carbonyl. There are two general ways to do this:
1. Add 1 equivalent of trifluoroacetic anhydride to the desired carboxylic acid, stir for a while (a few minutes was sufficient when using TFAA/acetic acid) and you have an almost quantitative yield of mixed anhydride which is stable to disproportionation. If necessary (as will be in your case) the TFA byproduct can be precipitated by treatment with an equivalent of pyridine or other tertiary amine.
2. Add 1 equivalent of trifluoroacetic anhydride to an equivalent of the plain carboxylic acid anhydride. The equilibrium shifts almost entirely to the formation of two moles of mixed anhydride, with no TFA byproduct.
The mixed anhydride which forms in both cases is a powerful acylating agent in its own right, and can acylate highly activated arenes with no catalyst. With a little extra TFA in the mixture as catalyst even normal aromatics (anisole, benzodioxole) can be acylated in 80-90% yield.
There is a comprehensive study on formation, stability and subsequent reactions of various mixed carboxylic acid-trifluoroacetic acid anhydrides in J. Chem. Soc., 1954, 2006-2011.* It details the Friedel-Crafts acylation of anisole I mentioned above, and also formations of esters and amides from mixed anhydrides. Although reactions of 3-phenylpropionic trifluoroacetic anhydride are not specifically detailed, the formation and reactivity of the mixed anhydrides is generic to all the 'normal' carboxylic acids, and nucleophilic attack occurs on the desired carbonyl (as long there is a large enough difference between the pKa values of the respective acids, but this is not a concern in your case).
*One of the co-authors of this paper was J. M Tedder, who was also the author of Chem. Rev. 55, 787-827 (1955) (https://www.thevespiary.org/rhodium/Rhodium/pdf/trifluoroacetic.anhydride.pdf)
(https://www.thevespiary.org/rhodium/Rhodium/pdf/trifluoroacetic.anhydride.pdf).
I'm curious as to why you want the N-benzyl amide. My first though was a Hoffman reaction to the PEA but I don't know why you'd want a benzyl group around if that was your intention. Is it yet another top-secret project? ;)