I have done the ‘sustrans c2c’ 5 times now (including all the off road bits, old coach rd out of keswick etc etc) and have used the same tyres each time – specialized crossroads. Every time I think,nah there must be something more narly for the off road bits, but each time I thank my lucky stars that I used them. Exceptionally quick on the road and old railway line stuff, and my set are the ‘old’ crossroads that have a bit more of a nobble of the side, so are ok on the off road sections.
I have recommended these tyres to each rider who has done it and we still haven’t had a flat yet.
In my view only, there isn’t a ‘perfect’ tyre for this sort of thing, BUT for the time you save and the ease of the road sections on this type of tyre you’ll enjoy the whole ride so much more.
Otherwise get a set of pan razers (or such like), blow them up to the max for the road and let some out for the off road.