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 I wrote a bit about this [in an unpublished article][1].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" (or maybe the name came from Mostow, who also wrote on it). 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