When I had discussed the opposite situation (smaller observed MOA groups at longer distance than at shorter) with Bill Davis Jr in the pre-acoustic target days, he suggested shooting through onion paper at shorter range to avoid altering the downrange path.
Franklin Mann in his 1909 book "The Bullets Flight from Powder to Target, used this method to document bullet's paths as they travelled down range.
Today you can use acoustic targets that triangulate the bullets position as it passes through the target plane without having to touch paper. Here's one example.
https://oehler-research.com/system-86-acoustic-target/
