Just in case others have the same odd policy as Barnet Council: payment by credit card (no AmEx, of course) 1.8% surcharge, payment by Visa and MC debit card no surcharge.
See what's missing? Yes, Visa Electron, which apparently includes the PayPal Top Up Visa card. OK for HMRC and other official bodies but not for Barnet Council.
A phone call establishes that no, they don't accept Visa Electron online, but yes, they do accept it by phone. So I do (indirectly) earn miles on my council tax payment after all.
If your council is as antediluvian as Barnet, it may be worth giving them a call.