"Euclidean QG" developed by hawking and friends in 1980s was a path integral
and so it would be closer akin to Renate Loll Lorentzian path integral by CDT method ("causal dynamical triangulations") that we hear a lot about these days
Hawking never got Euclidean path integral to work, but he uses it to think with. It sounds a bit eccentric for him to call it the "only sane way to do nonperturbative QG"
the Lorentzian path integral people (Loll et al) have an equally nonperturbative approach that they are getting results with, including confirming a conjecture or two of hawking. No way is Loll's approach not sane. It is at least as sane as the Euclidean version.
I need to get you some online links. there is a 1998 survey of QG methods by rovelli which describes hawking Euclid. path integral. More recent online stuff do not discuss hawking's method very much because it is long obsolete except for him and one or two proteges. But I will get the link to the 1998 survey
Strings, loops and others: a critical survey of the present approaches to quantum gravity
Plenary lecture on quantum gravity at the GR15 conference, Pune, India
"I review the present theoretical attempts to understand the quantum properties of spacetime. In particular, I illustrate the main achievements and the main difficulties in: string theory, loop quantum gravity, discrete quantum gravity (Regge calculus, dynamical triangulations and simplicial models), Euclidean quantum gravity
, perturbative quantum gravity, quantum field theory on curved spacetime, noncommutative geometry, null surfaces, topological quantum field theories and spin foam models. I also briefly review several recent advances in understanding black hole entropy and attempt a critical discussion of our present understanding of quantum spacetime."