Unfortunately, things are subtle and terrible (at least, compared to the abelian case). I wrote a bit about this [in an unpublished article][1]. 

One of the surprising, terrible features is that you can have two connections ω,η such that (1) curv ω = curv η, yet (2) ω and η are *not* gauge equivalent. [Wu and Yang][2] discussed this problem, and I think also were responsible for calling it the "field copy problem". It is quite weird when you sit down and think about it in terms of Yang-Mills field theory. From that perspective, you're looking at a connection whose dynamics is determined by a Lagrangian that only involves the curvature. The field copy problem shows that with nonabelian gauge groups, you can have two *distinct* connections that minimize the same Yang-Mills functional. In other words, observing the curvature form doesn't tell you enough to reconstruct the connection up to gauge equivalence, even in the special case of Yang-Mills connections! 

Len Gross analyzed the field copy problem in ["A Poincaré Lemma for Connection Forms"][3], where he found an alternative collection of observables that suffices to reconstruct the connection up to gauge equivalence. I tried to make sense of his results using 2-categories in the article I linked above. Ultimately my research went in a different direction and I never got to follow through with this thread in the way I would have liked.

There is also an issue with the next term of the exact-ish sequence, too (your $d_\omega$): the Bianchi identity sure *looks* like it should be the map to use there, as you propose. Ok, so we'll differentiate the 2-form with respect to the connection... but wait! We don't even *have* a connection form to put our hands on at that point! There is no $\omega$ to use!

  [1]: http://pi.math.cornell.edu/~noonan/halifax.pdf
  [2]: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.12.3843
  [3]: https://pdf.sciencedirectassets.com/272601/1-s2.0-S0022123600X04636/1-s2.0-0022123685900965/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEBUaCXVzLWVhc3QtMSJHMEUCIANT54Y31WoS6AOk4AgUxTD%2By1sLVSg4eAWz5f0gx1g2AiEA7%2BWf3jMjI%2FapHLD4Dr%2F2UAS%2FV8fBT6A5gaV0C4j1cOYqvQMIzv%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgwwNTkwMDM1NDY4NjUiDF0aKwdurwQwYQ%2FokCqRAzUZRP%2FKudHckFCyiMb8FCo5SzDYy1htB%2Brz40s0eQM9huXf3arVrpZyWYMeTSXmAEG%2Fcax7LcacxCpsUAk0clmOU2v64iceVuiKlD3%2BJJ%2B%2B4WROMxUxHhGCPRKVuKNANnPHBA1SrUgh75sYU5%2B9UV3iXi5AyXOkMn1jJwKHP9j1KPXtRK0rmq8XrNPGJ1%2F34x97wnUqCn1kf6uIm9ttC4yYolWDG4PQp5fbur%2FIGDPjsJGPjnxd0d2bOmNj6BK30gtfP7TsM%2BX0oZIXaeChcpbwmWer%2F3Z5IJcSDoucU5c6akwNiOBiqL%2FQC4Px2ofh%2F7TpBMrzCoqy86sKNl%2BTiEKIgNyijrYBQgOUp8yZdXfLVVCKFbEb3i47KW%2FheSNE0er3bvXRRtgK9m8wmEvNx9ZuZfYb4IdCMTyjvAOySMjm0X6dEg133zYR6zC%2F27oVYiCkkb1NvT7%2F1gkyxEgo9Zq40JXe4N%2BgozPkPQqoaGbHAJPxF8v7QvNwthHXz4oDPZXleztZjQGhdJ0OXf4PUFkmMPKdxf8FOusBKw2vmz4JoD6COrZOQOcauufrg2UGFfBYNfZlaGhjjxNZsih9mOTMgdAw868SZHFVp18qpVUW27dFfJATH6gQCCqZRB28Bu44GeWyxp1zlKuNtKW5VZruj8lJCeRfi1CZl1ctx%2BXJdlntmBZjFJUm7tHgeAz5%2Fz0GDLtcJPQ1ca%2FgiiyQYFvIVtCd0V3HkZwVjf5ETjApgOjix5WZBt7Ozhx7BAui1qpNjhbk6%2FJyIQo%2FxMOCIJ2BshwtKc1TOFVNGf%2B7YhH4oVi4uS779ne6ykhYvulTv%2BCcKAb7Z1q3nxJGOfHDI0TLB%2Ftjxg%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20210103T053959Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTY6B6ZF7P4%2F20210103%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=674847accb7d594b8696f2bcca2ccf0e7c535d6e2635915a3ee0f3eea9344e52&hash=788ea5f68fe00626ddd5f7ee19d532d012b6fcf98fab6ec4c2a83151313ee111&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=0022123685900965&tid=spdf-f04d116c-2857-4675-a59f-1ff3bedd7aec&sid=20d18fd277d2d1493c09ea2312af5b244ce0gxrqa&type=client