Please explain what evidence a general partnership should provide when opening a bank account to be CIP compliant.
Distinguish between entity registration and DBA registration, they are not the same thing. In general, only creatures of statute; e.g. corporations, LLCs, limited partnerships, etc. are required to register with the state in order to exist. On the other hand, virtually any business type that uses an assumed name may be required to register that name.
Supplementary information accompanying the final CIP regulation listed a partnership agreement as a sample of what you might require for documentary verification, but did not make it a requirement. (Some partnerships recognized as such by state law do not have written agreements.) As noted, any requirement to register a general partnership is a function of state law; e.g. Florida allows it, but does not require it. Also, as noted, state law may require a partnership doing business under an assumed name to register at the state or county level.
Your question simply cannot be answered here except with generalities. Your own institution's CIP, however, must answer the question directly. That's your primary resource.