I've walked up the Pike a couple of times with a dog (a big strong mongrel in the autumn of her years). I've also walked other high fells with my Jack Russel and, more recently, with a friend's Patterdale terrier. They all got on fine.
As to routes, the summit area of Scafell Pike is an extensive boulder field which you and the dog must cross to reach the top regardless of which route you've ascended.
It's not a matter of steepness and distance because IME any fit well-exercised dog can walk the few miles up the Pike easily. It's all to do with the roughness of the terrain underfoot.
Some dogs are hardier than others, some are more adept at steep ground, some more tolerant of scree and boulders.
My advice would be to try walking the dog on rocky going somewhere low and with easy escape to a road or to grass. See how Fido gets on with rough going before tackling the Pike. During and after the walk watch carefully for signs of lame-ness and examine the dog's pads carefully.
Don't do the walk in very hot weather and be sure to take plenty of water for the dog to drink.