I've been up to Burnt Island and you could get there with a kayak cart, but I cannot imagine that the six portages between Burnt Island and Source would work with a cart (I've not been on those). Typically, the routes marked cart trails work with a cart and a few of the portages closest to access points are so worn that they could work...the rest would be a challenge.
That's why I have two canoes and zero kayaks. Half the paddle, twice the man. Kick this guy to the curb!!! ;-)