No issues at all, done it quite a few times now since they changed from the old style payment tolls booths. You just drive Threw now and anpr camera’s record your reg pretty much like the congestion charge in london in the beginning you had to register but now it’s automated any scheme car or a car that the road tax is registered as disabled is exempt.
Recently my brother bought a used mb car and although he change road tax it still was on there system still as except he contacted dart charge and they said until the get notifcation they do not charge.
You can got to there page to pay the charge and if car is exempt https://www.gov.uk/pay-dartford-crossing-charge it will come up as so and will not allow you to pay.
I know this as on two recent trip to see my brother my reg was saying exempt even though i’d taken off my private and put the original plate back on the scheme car.
Then i got my brand new car off the scheme and had normal road tax while i waited for the new certificate to come and the private plate was on that and it would not allow me to pay.