The Full moon - A full moon is, to the camera, just like any other front lit object in bright sun. This has several consequences:
The sunny 16 rule says that proper daylight exposure for well lit subjects under a clear sky is to set the shutter speed to 1/ISO at f/16. This will display the Moon as it really is. Since it is a rock, the Moon will appear as roughly middle gray. The Moon normally appears to us to be quite bright so you need to adjust your exposure; overexpose by a stop or so. Thus we have the "Looney" (for Lunar) 11 rule: at f/11, shutter speed should be 1/ISO.
When the Moon is partially covered by clouds or is low in the sky you will need to increase your exposure. If you can't remember these rules just expose for something well lit in the foreground and, either in Photoshop or otherwise, increase the brightness of the Moon until it appears to be white. You should also bracket.
The full moon is a front lit subject. When shooting it you are in between the Sun and the Moon. The Earth is almost exactly between the Sun and Moon when there is a full moon (it is exactly between the sun and moon during a lunar eclipse). The full moon rises and sets 180 degrees from the Sun. So at sunset the Moon will be low in the eastern sky, and at sunrise, low in the western sky. On most days of the month you cannot see the full moon during the daylight hours. A waxing (increasing in size) nearly full moon can be seen low in the eastern sky just before sunset. A waning nearly full moon can be seen setting low in the western sky at sunrise and will not be visible at sunset.
The sunrise/sunset tables on this site include Moon phase and moonrise/moonset times for selected locations. You can visit sunrisesunset.com to get times for anywhere in the world. You can get the azimuth (direction), altitude, and percent illuminated of the Moon at http://aa.usno.navy.mil/data/docs/AltAz.php. The same information is available using an application called The Photographer's Ephemeris (TPE) which is available on Windows, Mac, IOS, and Android. See photoephemeris.com.
There is a lot of good information here. Note that I have assumed you would shoot about 25 minutes after sunrise at 6:26. At 6:26 the Sun should be about 3.9 degrees above the horizon so the arch should be lit. I haven't been to Top Rock Arch recently but I expect that it would be lit even earlier since it is at a high point. The thick yellow line on the image is the Sun's direction at sunrise; the thin yellow line is the Sun's direction at 6.26 AM.
At 6:26 the Moon will be 6.4 degrees above the horizon and is in the direction of the thin dark gray line which goes through The Notch. By moving the small gray marker you can get the apparent elevation of anything that might come between you and the Moon. I moved the gray marker to the area near The Notch and slightly varied its location until I reached the highest elevation. From the "Geodetics" section of TPE you can see that the The Notch is a bit higher than Top Rock Arch, about +75 m. The apparent altitude is +3.0 degrees. Since the Moon's altitude is 6.4 degrees above the horizon at 6:26AM, the Moon should be about 3.4 degrees above The Notch and you should be able to shoot it though Top Rock Arch.
Alternatively, and less precisely, you could go to http://aa.usno.navy.mil/data/docs/AltAz.php. to get the direction and altitude of the Moon during the hour before the Moon sets. Here's a sample of the output:From the table the Moon will be roughly WSW (240 degrees) when it sets, 95% illuminated, and within 10 degrees of the horizon. This method is less precise since it doesn't take into account the topography of the area. If The Notch was a bit higher it would be possible that it would block the Moon.
In practice there is a problem with both methods: local topography does not always appear on Google maps. Having been to Top Rock Arch, I know you cannot get very far from the arch when photographing it. A normal to short telephoto lens is the best you can do. As a result, the full moon will be very small inside the arch.