It may differ depending on the LA, but here they are separate functions, so notifying one will not have any bearing on the other. I may be wrong in my recollection but I'm not sure I needed to notify BC i had started, only when it came to the first inspection stage (foundations).
Just make sure both planning and BC acknowledge in writing. If you are really worried as it's so tight, maybe consider discharging the relevant planning condition (ie call them and pay the £100 fee)?