You'll certainly be banned from the upper section of the walk alongside Loch Lomond. Although quite easy to get a ferry across from Rowardennan to the West bank - and continue walking North to rejoin the path at Beinglas Farm, you & your dog will be walking alongside a very busy main road!
I've a strong feeling that there's another section where dogs are not allowed, but not sure where. The first time I walked it, I met a couple with a dog on the first 2 days, but our paths diverged after Rowardennan, so I don't know whether they made it any further.
In all honesty - it is not a dog friendly route (not due to terrain - but due to land-owners). I'm sure there's more info on the official web-site for it.