As a general principle, I never sign anything which does not require a signature. And just because a pilot pays to fly with an FI, he/she doesn't own that FI, so the view that being a 'customer' entitles you to have your logbook signed by the FI for any instructional flight is incorrect.
An FAA BFR would only count as a JAR-FCL training flight for the purposes of revalidation if conducted by an FI authorised under JAR-FCL, as I understand it.