Not sure what you mean by ‘monitor’. It’s your meter which records your usage at whatever time of day, not any other device. If you’ve got an E7 meter it will switch counters once a day, if it’s E10 it will switch three times. The timings will probably be hard coded into the meter but my be controlled by radio signal. Your supplier will either add both readings together and charge you a single price, or it will charge you a lower rate for the off-peak reading. If you have an E10 meter and it has a specific E10 tariff it will most likely charge you that. If it only has one time-of use- tariff, probably called E7 but it doesn’t matter what it’s called, you will pay that no matter how many times your meter switches counters each day. That assuming they will supply you with an E10 meter.
But, and it’s a big but, if you have a smart meter you need to know exactly what it is set up to do, and only your supplier will know that. And remember, they can change what it does remotely any time they like.