In Photoshop, I first combined both images, then resized and repositioned the moon so it overlapped the fuzzy moon in the original image. I didn't want the moon so large that it obscured some of the interesting cloud/haze/halo that the original shot provided.
I thought at first I would be smart and fancy, and use a Blend Mode to blend the two shots together, but my attempts with that technique weren't to my liking, and likely are due to my basic experience using blend modes to begin with.
Instead I opted for masking. I used the Eliptical Marquee to select the moon, and then inverted the selection. I did not want a crisp circular cutout, so I first expanded the selection by 4 pixels (Select > Modify > Expand) and then feathered the selection by 2 pixels (Select > Modify > Feather).
Lastly, with the moon layer selected, I applied some uniform, monochromatic noise, so that the moon more closely matched the noise/grain of the background image.