Try to use Channels to cut out the fire:
Select the picture layer of the fire, and click channels ( if it's not active near the layer tab , go to Window -channels)
Choose the channel with most contrast and use select color range to select the fire. Use the fuziness bar to adjust the selection. after you click ok and have the selection press Ctrl J to make a new layer out of the selection. Try it multiple times until you like what you get.

Selectiv via channel gives you a more transparent look which blends better in background.

Thank you so much.