I'm sure heads at independent schools will not get petty about non-attendance if there is a reason (important visitors, pointless lessons etc). Why should they? I'm not sure they have the same powers to fine as state heads anyway.
State heads don't have the power to fine either. All they can do is report you for non-attendance. Authorities outsource some of their obligations (as child protection, not as education authorities) to state schools, but the power to fine is a magistrate's, and arises from the criminal offence of not ensuring effective education. As to whether an independent school would inform the authorities of a non-attending child, who knows? One would hope they would, and one stick might be that a local authority that believed a school was concealing persistent non-attendance could get very nasty: they could, for example, bring "not providing effective education" actions against all the parents. Independent schools I've looked at all had a parental undertaking about no unauthorised absences, but what their sanctions are, who knows?
But it seems a little perverse to pay for a school which offers pointless lessons you then withdraw your child from (I'm going to go to a restaurant I don't like, order food I won't eat and walk out: that'll show 'em!)