If I start walking the Camino del Norte at the beginning of September, is booking a place to sleep useful and/or necessary in advance?
This is one of those questions that can only be answered by opinions. And opinions will vary..
What is pretty clear is that early September is a very popular time to start the camino from Irun.
So most pilgrim accommodation that is available then is likely to be filled to capacity or near-capacity.
Reasonable costing pilgrim accommodation is likely to be booked well in advance - check a few samples on Gronze for places where you think you'd be likely to stay.
Not all accommodation is bookable in advance. But 'walk up' options are becoming quite limited on the Norte - so again, check for these on Gronze, look at bed numbers etc and decide if you think you are likely to arrive in good time ahead of others to snag a bed.
IMO if you walk at a really popular time you have to be prepared to do one or more things like - plan and book accommodation advance, get up sufficiently early and walk sufficiently fast to win the bed race, pay more for hotel accommodation (often expensive at the touristy locations on the Norte), focus on staying calm and not get stressed by any setbacks, such as other people taking accommodation you'd hoped to have for yourself! Being focused and calm might also mean spending a lot of time systematically going through accommodation options online and on apps, and might also mean taking a taxi to somewhere off the camino if you find there are no options around you and you don't want to spend the night out of doors.
The odds are that it will be very busy, but no-one knows for sure - 'pilgrim bubbles' or days where there are few pilgrims walking occur naturally and no-one really knows when this will happen - except that weekends and Mondays are often the busiest days for people to begin from an accepted start point. So starting mid-week could be a helpful strategy.