'The inspector and the engineer disagree — who wins?'

Neither, formally. The Building Control body's interpretation of the Regulations is what counts on site. If the engineer believes the inspector is wrong, get it in writing and ask for a manager-level review.

'Can we proceed without a stage sign-off?'

Technically yes, with risk. Practically, do not. Cover-up means uncovering if the next inspection picks something up.

'The client wants to skip an inspection — what do I do?'

Document the request, decline politely, and explain the consequences. Your name is on the work.