Is it possible to react two distinct carboxylic acids to form a dicarbonyl ester?
The standard term for such molecules is acid anhydride, as they can be viewed as the product of a condensation reaction between two carboxylic acids, with concomitant loss of $\ce{H2O}$. Asymmetric anhydrides (i.e., ones that decompose into two distinct carboxylic acids if hydrolyzed) can certainly be prepared. One possible route is to synthesize an acid chloride from carboxylic acid #1 (thionyl chloride, $\ce{SOCl2}$, being one common reagent used to make the acid chloride) and then react it with the carboxylate salt of carboxylic acid #2. In that reaction, the carboxylate functions as a nucleophile and attacks the carbonyl carbon of the acid chloride, with chloride anion expelled as the leaving group.

I don't think it's an efficient laboratory method to prepare asymmetric anhydrides by simple dehydration of stoichiometric mixtures of distinct carboxylic acids. I suspect that would probably yield a (roughly) statistical distribution of products, at least in the ideal case. In reality, various factors might skew the distribution one way or another.