On a Falcon 9 booster, does the central engine experience unique variations of its ISP due to its plume bordering the eight other plumes? If there is any influence, how does the ambient relative pressure evolve along the whole burn between lift off and main engine cutoff, in comparison with the other engines of the cluster?
It is impossible that there is any influence, upon any of the engines, from the plumes bordering on each other. Any perturbation in a gas can only travel at the speed of sound in that gas; as the exhaust velocities are a multiple of the speed of sound in the exhaust gas, a perturbation can never travel back into the engine. See the classic by Huzel and Huang, "Modern Engineering for the Desing of Liquid Propellant Rocket Engines", chapter 2.