I currently bank with Monzo and it’s using a debit MasterCard.
I haven’t yet contacted TfL or Monzo regarding the double transactions because I know from past experience that they have no interest unless the transaction actually goes through – so all the time it’s pending they won’t do anything.
My previous issues were when I used Curve and NatWest cards and in every occasion bar one it was with an early morning journey.
I would find that the first bus trip was fine but if TfL chose to do the 10p authorisation while the bank was doing it’s daily maintenance then the second bus trip would fail because the NatWest would frequently decline the 10p authorisation during its downtime – this nearly got me stranded in Welling at 4.30am on a few occasions as my card was now not accepted for travel!
This was the main reason that I went back to using Oyster despite TfL pushing the use of contactless.