Counterexample:

  ...XX
  SF.FD
  ...XX
  
  S = start
  F = fire
  X = wall
  D = destination

The cat can to the destination in 6 moves passing through 1 fire. In the fireWillingness=1 pass, the middle tile is reached after passing through fire, so the destination appears unreachable. The proposed algorithm will pass through 2 fires instead of 1.