Is it possible to build a compact proof of the sum of all even numbers of a dataset? Suppose I have a set of numbers:

[1, 7, 6, 4, 8, 9, 4, 6, 3, 3, 1, 5, 4]


Suppose also that a verifier knows the merkle root of that set, R, but he doesn't know the set itself. Is it possible to convince him that the sum of all even numbers - i.e., 6 + 4 + 8 + 4 + 6 + 4 is 32 with a string of a small, constant size (i.e., not depending on the length of the array)? 

I thought that was possible with zk-snarks, but now I'm under the impression even a zk-snark proof would be of linear size w.r.t the number of elements.
Let's ignore the zk requirement. And focus on compactness. If we have a merkel tree and I swap just one element at random so one of the hashes is invalid and doesn't match the data. We get a merkel tree with almost all valid hashes. Someone would have to verify all of them to catch the foegery. If you only sampled a few paths you would gain little confidence.
Any extra calculations you require on the tree could be done onhe forged version. I'm afraid your request is impossible.