I would check again with the agency but I would expect only 45 mins to be deducted for all breaks regardless of what the tacho says. So therefore you would be fine to leave it on break. You cannot remove the card without a good reason, so that’s not an option…
The theory is that you put it on POA, but that will mean the agency will lose out having to pay you for all the time sat doing nothing. I’m sure they’ll be reasonable and only deduct 45 mins.
Where I was, even if driving time was over 9 hrs, so more than one 45 min break was required, you would only be deducted one 45 min, because the other one was only because you were doing overtime.
Worst case, they can email you a timesheet for you to printout beforehand, or they post them to you (no use for Saturday), or the client has a stock of them if they use that agency a lot? Agent should have given you a supply when you registered with them!