If my understanding is correct, FAR 23/25 is only for airworthiness standards and not operating legislation, such as Part 91, 121 and 135. Basically how I understand it is in order to achieve the performance they achieved during certification, I need to follow the procedures laid down in FAR 23 or 25, whichever one applies.
FAR 23/25 may be for certification standards, but those standards are designed to mesh with TERPS criteria, and airlines' OpSpecs are designed to provide operating criteria to meet AT LEAST the basic requirements.
If an airline's OpSpecs and training program are approved to meet higher criteria (e.g., the 800' min accel height instead of the 400' basic regulatory min), then you are obligated to operate in accordance with those rules and training. There should NEVER be a case where the OpSpecs or Training Manual has looser criteria than regulatory minimums, except if a specific waiver is in force.