What bank accounts are UK charities using?

We (CEEALAR) are currently looking at switching banks because of some issues we are having with staff access, payment verification, and bank cards not working (Mastercard debit cards are really bad for this it seems, to the point of being unusable for online ordering). Ideally we want a set up such that payments over a certain amount require verification by a second user. This feature doesn't work with our current bank as a single user can put a payment through on their laptop, and then verify the same payment using their phone, thus totally defeating the point of this security feature!