Totally fine I've been riding around the Eastern half of the south downs recently on 33m spesh tracers tubeless and leaving my ht 29er mtb at home as it isn't necessary
I'd say the middle section is probably least narrow tyre friendly - personally I wouldn't be that keen on doing the whole thing on anything under 40c. But it very much depends on your tolerance for fixing punctures/discomfort/dodging larger rocks.
Totally fine I've been riding around the Eastern half of the south downs recently on 33m spesh tracers tubeless and leaving my ht 29er mtb at home as it isn't necessary