The refurb place will probably charge £10 extra to remove the old tyres and refit (old or new) ones once refurbed, where as the Blackcircles offer is just £1 on top each when buying the tyres.
I personally would take the wheels away and let them fully cure for a couple of days before having the tyres fitted anyway but that's just me.
I suppose the one advantage of having the refurb place fit them is as they have just painted/powder coated them they 'might' take more care not to mark them when fitting them but probably aren't quite as competent as a 'good' tyre fitter if you can get one