I am not a particular fan of any one insurer and I will always take up an issue direct with an insurer if i think it is a fair point that has been raised. However, I do not think that you understand the Barclays offer. They say that they will match your RENEWAL, not any old quote that you can muster from another source.
In order to qualify for the payment, they wish to see the renewal papers from your EXISTING insurer. They are not offering to match the price offered by other insurers and indeed they would obviously match your renewal as they gave it!