I want an offensive tactical coach leaving Lambert out of the equation for me IMO. Sad news though that he lost his wife to the age of 45, if I browsed through that story correctly.
Penguins won the the Cup using their strengths to their advantage and Chicago has won multiple Cups being on the offensive side of the spectrum (though with smart decision making). That Penguins team also had a defensive core of Letang, Daley, Maata, Lovejoy, Schultz, Cole, and Dumoulin. True Avs don't have a defender who can match Letang in terms of talent, but Avs easily have a better top 3 now and have a better top 6 in terms of potential. I think Avs can win the Cup with the defensive core they have now, but with better coaching tactics which doesn't need to focus on boring defensive structures, but smarter decisions with the puck creating offensive chances with good puck possession. Look at the Dallas Stars and all they went through with a poor defensive core and just horrid goaltending, but won the division due to playing offensive majority of the time. Avs don't need a Jacques Martin because of how bad defensively the team is, what they need is a coach to realize the strengths of this club and play in a structured way utilizing the speed, size, and possession ability of these players IMO.
And because of that Bednar and Green are my top two candidates.