I was wondering if it would be feasible to do Iceland's route 1 but only booking accommodation one or two nights in advance? Specially looking at late September, and only requirement of accommodation is a private ensuite.
Yes, it should work just fine.
Sadly most tourist come at the normal tourist season, mid-June through August, which means they missout on a lot of Iceland. However this is good for you as you'll be traveling during mid-low season.
Since there are fewer visitors there are more free rooms which would mean that your plan should work for sure.