Proving multiple products "in the exponent"
Unfortunately, improving over the $O(n)$ solution seems to require the use of non-standard assumption. More precisely:


No, Groth-Sahai NIZKs cannot be used for proving this statement with communication less than $O(n)$, because the witness held by the prover is already of size $O(n)$. In fact, even if the prover knew all the discrete logarithms of the group elements (i.e., all the $a_i$'s), Groth-Sahai proofs would still produce proofs of size $O(n)$. The main reason for this is that Groth-Sahai proofs are... Well, proofs (more precisely, they can be either proofs or arguments, depending of the mode of the crs, but they have the same size in both cases since the modes must be indistinguishable). An interactive proof (meaning, with statistical soundness - as opposed to interactive arguments which have computational soundness), zero-knowledge or not, non-interactive or not, cannot have communication sublinear in the witness size unless NP can be decided in subexponential time.
It does not seem to be straightforward to obtain a shorter proof for your problem without using "heavy hammer" tools, or the random oracle model. If you are ok with using the random oracle model, then there are sublinear interactive proofs which can prove this statement with logarithmic communication complexity, and can be made non-interactive using Fiat-Shamir (1, 2). In fact, very recently, it was shown here how to build such succinct interactive public-coin arguments directly for bilinear equations a la Groth-Sahai. If you do not want to use random oracles, then you will have to rely on the non-standard and non-falsifiable assumptions underlying the constructions of all SNARGs - there are many available. However, none of them will provide a very good concrete efficiency for your specific problem, and I'd strongly suggest to rely on a solution in the random oracle model instead.