Via email, Noah Snyder pointed me to the papers https://arxiv.org/abs/math/0301142 and https://arxiv.org/abs/1102.3221. I wound up going in a different direction than this question was suggesting, so I did not read the papers in detail and try to translate their results to more diagrammatic language, but it seems like that should be possible.

In particular, Section 7.9 of the first paper (by Tuba and Wenzl) has the type of recursive formula I was looking for.

Via email, Noah Snyder pointed me to the papers Tuba and Wenzl - On braided tensor categories of type BCD and Lehrer and Zhang - The second fundamental theorem of invariant theory for the orthogonal group. I wound up going in a different direction than this question was suggesting, so I did not read the papers in detail and try to translate their results to more diagrammatic language, but it seems like that should be possible.

In particular, Section 7.9 of the first paper (by Tuba and Wenzl) has the type of recursive formula I was looking for.