We did the Way of the Roses in September 2013; you can read about it (and the return trip)
here. Surfaces are pretty good - it's all tarmac - and there's nothing that would challenge someone B&Bing on a road bike with 23mm tyres. But....
You say you'll be camping. I'd have to say I'd be thinking about wider tyres. And hauling a camping load up and down steep hills (unless you and the load are both
serious lightweights) is asking for trouble on racing wheels (32 spokes or less). It might be worth using this as an excuse to get a tougher bike or maybe just a pair of tougher wheels with wider rims, 36 spokes and 32mm tyres. At least, consider putting a 32 mm tyre on the back - this will have the volume to cushion the spokes from any serious potholes and might save you a busted spoke or two.
Yeah, Derrick did it on a road bike - but he wasn't camping. Unless you're a real gnarly minimalist camper, camping gear tends to add up to a considerable weight. Hauling it up a hill adds further strain to those rear-wheel spokes...
There are some challenging climbs as you come through the the Dales. Pately Bridge and Settle areas in particular: be prepared for those with strong legs or low gears or both, and check your brakes - especially if you're hauling a load.
Yeah, our tyres are a bit more than 23mm. But then there are two of us on the bike - and our camping gear.
PS - is it my ageing eyesight, or is there no rear brake on your bike?