Yes, I agree with that. In this context, "planning" means a person from the local council (i.e. local government) who give permission for building work to go ahead. I suspect, from what they say, that these people are working on a listed building, where you have to get "planning permission" to do anything significant to the building.

I think I recognise this as an episode of Grand Designs, right? "...you'd would've had planning round..." - but it sounds more like "you'd've had planning round..." because of his natural speech. So in this context I think he's referring to having people come to the house - they are 'coming round' e.g. 'My family are coming round later.' - and these people are involved with the planning and architecture. People involved with planning are visiting the house = having 'planing round'.