It's signposted. You get to a point where it tell you - Sutton Bank unsuitable for caravans please follow the alternative route.
I don't think they would give the alternative route if it wasn't suitable.
My friend goes to that area regularly with his caravan (from Cumbria) and he's never had a problem following it.
We came down that road a few weeks ago and its not the best road. It is obviously less miles but still very steep and narrow. May be just as easy heading to York then doubling back.
We recently went on the alternative route for caravans from the west avoiding Sutton Bank - our caravan is 24' in length and single axled.
Our choice of routes to get to Pickering were the longer way round via the York Ring Road and onto the A64 (from experience it's a car park a lot of the time) or take the signed official diversion.
We had no problems whatsoever, yes, it's narrow in a couple of places with poor visibilty around the odd bend or two,it's also got the odd hill (let's face it it is the North York moors and Howardian Hills after all!)and going through Ampleforth village you would need to look well ahead and not commit until you can see it is clear, but all in all it's relatively easy paced and pretty drive.
I figure if you've the confidence, ability and probably most importantly the towing vehicle to manage your caravan this route offers just the most minor of challenges.
Can I just remind anyone travelling west to east (I see you're in Cumbria) that the A59 is still closed at Blubberhouses, with lengthy diversions in place. Officially going to be closed for the next couple of weeks (and could be longer than that!)
Edit : Just noticed date of original post! Sorry! Leaving my comment just in case it's of use to anyone.
we live in stockton on tees . our way is A19 then A174 then A171 Whitby Scarborough road . We use this road all the time and we had a 26ft caravan too .
We live stockton on tees and go to scarborough a lot . Our way is A19 then A174 then A171 the whitby /scarborough road . we had a 26ft caravan as well and it was no problem at all .